Fundamentals of Graph Theory

Graph Theory - Fundamentals

Fundamentals of Graph Theory N.K. S UDEV

Vidya Academy of Science & Technology Thalakkottukara, Thrissur-680501.

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N.K. Sudev Fundamentals of Graph Theory

Vidya Academy of Science & Technology


A graph G can be considered as an ordered triple (V, E, ψ), where I V = {v1 , v2 , v3 , . . .} is called the vertex set of G and the elements of V are called the vertices (or points or nodes);

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A graph G can be considered as an ordered triple (V, E, ψ), where I V = {v1 , v2 , v3 , . . .} is called the vertex set of G and the elements of V are called the vertices (or points or nodes); I E = {e1 , e2 , e3 , . . .} is the called the edge set of G and the elements of E are called edges (or lines or arcs); and

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A graph G can be considered as an ordered triple (V, E, ψ), where I V = {v1 , v2 , v3 , . . .} is called the vertex set of G and the elements of V are called the vertices (or points or nodes); I E = {e1 , e2 , e3 , . . .} is the called the edge set of G and the elements of E are called edges (or lines or arcs); and I ψ is called the adjacency relation, defined by ψ : E → V × V , which defines the association between each edge with the vertex pairs of G.

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A graph G can be considered as an ordered triple (V, E, ψ), where I V = {v1 , v2 , v3 , . . .} is called the vertex set of G and the elements of V are called the vertices (or points or nodes); I E = {e1 , e2 , e3 , . . .} is the called the edge set of G and the elements of E are called edges (or lines or arcs); and I ψ is called the adjacency relation, defined by ψ : E → V × V , which defines the association between each edge with the vertex pairs of G. Usually the graph is denoted as G = (V, E). The vertex set and edge set of a graph G are also written as V (G) and E(G) respectively. December 1, 2017

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Figure: An example of a graph. December 1, 2017

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An edge of a graph that joins a node to itself is called loop or a self loop. That is, a loop is an edge uv, where u = v.

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An edge of a graph that joins a node to itself is called loop or a self loop. That is, a loop is an edge uv, where u = v. In the above example, the edges e6 and e7 are loops.

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An edge of a graph that joins a node to itself is called loop or a self loop. That is, a loop is an edge uv, where u = v. In the above example, the edges e6 and e7 are loops. The edges connecting the same pair of vertices are called multiple edges or parallel edges.

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An edge of a graph that joins a node to itself is called loop or a self loop. That is, a loop is an edge uv, where u = v. In the above example, the edges e6 and e7 are loops. The edges connecting the same pair of vertices are called multiple edges or parallel edges. In the above example, the edges e4 and e5 are parallel edges.

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An edge of a graph that joins a node to itself is called loop or a self loop. That is, a loop is an edge uv, where u = v. In the above example, the edges e6 and e7 are loops. The edges connecting the same pair of vertices are called multiple edges or parallel edges. In the above example, the edges e4 and e5 are parallel edges. A graph G which does not have loops or parallel edges is called a simple graph. A graph which is not simple is generally called a multigraph.

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If two vertices u and v are the (two) end points of an edge e, then we represent this edge by uv or vu.

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If two vertices u and v are the (two) end points of an edge e, then we represent this edge by uv or vu. If e = uv is an edge of a graph G, then we say that u and v are adjacent vertices in G and that e joins u and v. (We may also say that each that of u and v is adjacent to the other).

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If two vertices u and v are the (two) end points of an edge e, then we represent this edge by uv or vu. If e = uv is an edge of a graph G, then we say that u and v are adjacent vertices in G and that e joins u and v. (We may also say that each that of u and v is adjacent to the other). Given an edge e = uv, the vertex u and an edge e are said to be incident with each other and so are v and e.

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If two vertices u and v are the (two) end points of an edge e, then we represent this edge by uv or vu. If e = uv is an edge of a graph G, then we say that u and v are adjacent vertices in G and that e joins u and v. (We may also say that each that of u and v is adjacent to the other). Given an edge e = uv, the vertex u and an edge e are said to be incident with each other and so are v and e. Two edges ei and ej are said to be adjacent edges if they are incident with a common vertex.

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The order of a graph G, denoted by ν(G), is the number of its vertices and the size of G, denoted by (G), is the number of its edges.

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The order of a graph G, denoted by ν(G), is the number of its vertices and the size of G, denoted by (G), is the number of its edges. A graph with p-vertices and q-edges is called a (p, q)graph.

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The order of a graph G, denoted by ν(G), is the number of its vertices and the size of G, denoted by (G), is the number of its edges. A graph with p-vertices and q-edges is called a (p, q)graph. The (1, 0)-graph (a graph with a single vertex) is called a trivial graph. An empty graph is a (p, 0)-graph (a graph without edges).

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The order of a graph G, denoted by ν(G), is the number of its vertices and the size of G, denoted by (G), is the number of its edges. A graph with p-vertices and q-edges is called a (p, q)graph. The (1, 0)-graph (a graph with a single vertex) is called a trivial graph. An empty graph is a (p, 0)-graph (a graph without edges). A graph with finite number of vertices as well as a finite number of edges is called a finite graph. Otherwise, it is an infinite graph.

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The number of edges incident on a vertex v, with self-loops counted twice, is called the degree of the vertex v and is denoted by degG (v) or deg(v) or simply d(v).

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The number of edges incident on a vertex v, with self-loops counted twice, is called the degree of the vertex v and is denoted by degG (v) or deg(v) or simply d(v). A vertex having no incident edge is called an isolated vertex. In other words, isolated vertices are those with zero degree.

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The number of edges incident on a vertex v, with self-loops counted twice, is called the degree of the vertex v and is denoted by degG (v) or deg(v) or simply d(v). A vertex having no incident edge is called an isolated vertex. In other words, isolated vertices are those with zero degree. A vertex of degree 1, is called a pendent vertex or an end vertex.

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The number of edges incident on a vertex v, with self-loops counted twice, is called the degree of the vertex v and is denoted by degG (v) or deg(v) or simply d(v). A vertex having no incident edge is called an isolated vertex. In other words, isolated vertices are those with zero degree. A vertex of degree 1, is called a pendent vertex or an end vertex. A vertex with degree greater than or equal to 2 is called an internal vertex or an intermediate vertex.

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The maximum degree of a graph G, denoted by ∆(G), is defined to be ∆(G) = max{d(v) : v ∈ V (G)}.

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The maximum degree of a graph G, denoted by ∆(G), is defined to be ∆(G) = max{d(v) : v ∈ V (G)}. Similarly, the minimum degree of a graph G, denoted by δ(G), is defined to be δ(G) = min{d(v) : v ∈ V (G)}

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The maximum degree of a graph G, denoted by ∆(G), is defined to be ∆(G) = max{d(v) : v ∈ V (G)}. Similarly, the minimum degree of a graph G, denoted by δ(G), is defined to be δ(G) = min{d(v) : v ∈ V (G)} The degree sequence of a graph of order n is the n-term sequence (usually written in descending order) of the vertex degrees.

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The maximum degree of a graph G, denoted by ∆(G), is defined to be ∆(G) = max{d(v) : v ∈ V (G)}. Similarly, the minimum degree of a graph G, denoted by δ(G), is defined to be δ(G) = min{d(v) : v ∈ V (G)} The degree sequence of a graph of order n is the n-term sequence (usually written in descending order) of the vertex degrees. An integer sequence S is said to be graphical if there is a graph G with S as its degree sequence.

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In the above figure, δ(G) = 2, ∆(G) = 5 and the degree sequence of G is (5, 4, 3, 2).

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I

In a graph G, the sum of the degrees of the vertices is equal to twice P the number of edges. That is, d(v) = 2. (This result is known as v∈V (G)

the first theorem on graph theory and hand shaking lemma).

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the first theorem on graph theory and hand shaking lemma). I

For any graph G, δ(G) ≤

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2|E| |V |

≤ ∆(G).


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the first theorem on graph theory and hand shaking lemma). 2|E| |V |

I

For any graph G, δ(G) ≤

I

For any graph G, the number of odd degree vertices is always even.

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≤ ∆(G).


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A graph G is said to be a regular graph if all its vertices have the same degree. A graph G is said to be a k-regular graph if d(v) = k for all v ∈ V (G).

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A graph G is said to be a regular graph if all its vertices have the same degree. A graph G is said to be a k-regular graph if d(v) = k for all v ∈ V (G).

(a) A graph

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(b) A graph

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A directed graph or digraph G consists of a set V of vertices and a set E of edges such that e ∈ E is associated with an ordered pair of vertices. In other words, if each edge of the graph G has a direction then the graph is called directed graph.

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A directed graph or digraph G consists of a set V of vertices and a set E of edges such that e ∈ E is associated with an ordered pair of vertices. In other words, if each edge of the graph G has a direction then the graph is called directed graph. If we assign directions to the edges of a given graph, then the new directed graph D is called an orientation of G.

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A directed graph or digraph G consists of a set V of vertices and a set E of edges such that e ∈ E is associated with an ordered pair of vertices. In other words, if each edge of the graph G has a direction then the graph is called directed graph. If we assign directions to the edges of a given graph, then the new directed graph D is called an orientation of G. Note that an undirected graph G can have 2|E(G)| distinct orientations.

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A directed graph or digraph G consists of a set V of vertices and a set E of edges such that e ∈ E is associated with an ordered pair of vertices. In other words, if each edge of the graph G has a direction then the graph is called directed graph. If we assign directions to the edges of a given graph, then the new directed graph D is called an orientation of G. Note that an undirected graph G can have 2|E(G)| distinct orientations. If remove the directions of the edges of a directed graph D, then the reduced graph G is called the underlying graph of D.

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A directed graph or digraph G consists of a set V of vertices and a set E of edges such that e ∈ E is associated with an ordered pair of vertices. In other words, if each edge of the graph G has a direction then the graph is called directed graph. If we assign directions to the edges of a given graph, then the new directed graph D is called an orientation of G. Note that an undirected graph G can have 2|E(G)| distinct orientations. If remove the directions of the edges of a directed graph D, then the reduced graph G is called the underlying graph of D. Note that a directed graph can have a unique underlying graph. December 1, 2017

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In a directed graph D, the out-degree of a vertex v of DG, denoted by d+ (v), is the number of edges beginning at v and the in-degree of v, denoted by d− (v), is the number of edges ending at v.

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In a directed graph D, the out-degree of a vertex v of DG, denoted by d+ (v), is the number of edges beginning at v and the in-degree of v, denoted by d− (v), is the number of edges ending at v. The sum of the in-degree and out-degree of a vertex is called the total degree of the vertex.

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In a directed graph D, the out-degree of a vertex v of DG, denoted by d+ (v), is the number of edges beginning at v and the in-degree of v, denoted by d− (v), is the number of edges ending at v. The sum of the in-degree and out-degree of a vertex is called the total degree of the vertex. A vertex with zero in-degree is called a source and a vertex with zero out-degree is called a sink.

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In a directed graph D, the out-degree of a vertex v of DG, denoted by d+ (v), is the number of edges beginning at v and the in-degree of v, denoted by d− (v), is the number of edges ending at v. The sum of the in-degree and out-degree of a vertex is called the total degree of the vertex. A vertex with zero in-degree is called a source and a vertex with zero out-degree is called a sink. In a directed graph D, the sum of the in-degrees and the sum of out-degrees of areP equal to the number Pthe vertices of edges. That is, d+ (v) = d− (v) = . v∈V (D)

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The neighbourhood (or open neighbourhood) of a vertex v, denoted by N (v), is the set of vertices adjacent to v.

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The neighbourhood (or open neighbourhood) of a vertex v, denoted by N (v), is the set of vertices adjacent to v. Therefore, d(v) = |N (v)|.

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The neighbourhood (or open neighbourhood) of a vertex v, denoted by N (v), is the set of vertices adjacent to v. Therefore, d(v) = |N (v)|. The closed neighbourhood of a vertex v, denoted by N [v], is simply the set N (v) ∪ {v}.

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The neighbourhood (or open neighbourhood) of a vertex v, denoted by N (v), is the set of vertices adjacent to v. Therefore, d(v) = |N (v)|. The closed neighbourhood of a vertex v, denoted by N [v], is simply the set N (v) ∪ {v}. A graph H(V1 , E1 ) is said to be a subgraph of a graph G(V, E) if V1 ⊆ V and E1 ⊆ E.

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The neighbourhood (or open neighbourhood) of a vertex v, denoted by N (v), is the set of vertices adjacent to v. Therefore, d(v) = |N (v)|. The closed neighbourhood of a vertex v, denoted by N [v], is simply the set N (v) ∪ {v}. A graph H(V1 , E1 ) is said to be a subgraph of a graph G(V, E) if V1 ⊆ V and E1 ⊆ E. A graph H(V1 , E1 ) is said to be a spanning subgraph of a graph G(V, E) if V1 = V and E1 ⊆ E.

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Figure: Examples of Subgraphs

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A complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete graph on n vertices is denoted by Kn .

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A complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete graph on n vertices is denoted by Kn .

Figure: First few complete graphs

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edges (a triangular numA complete graph has n(n−1) 2 ber ), and is a regular graph of degree n − 1.

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Note: Every graph on n vertices is a (spanning subgraph) of the complete graph Kn .

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If the edges of a complete graph are each given an orientation, the resulting directed graph is called a tournament.

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If the edges of a complete graph are each given an orientation, the resulting directed graph is called a tournament. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction).

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A graph G is said to be a bipartite graph if its vertex set V can be partitioned in to two sets, say V1 and V2 , such that no two vertices in the same partition can be adjacent. Here, the pair (V1 , V2 ) is called the bipartition of G.

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A graph G is said to be a bipartite graph if its vertex set V can be partitioned in to two sets, say V1 and V2 , such that no two vertices in the same partition can be adjacent. Here, the pair (V1 , V2 ) is called the bipartition of G.

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A bipartite graph G is said to be a complete bipartite graph if every vertex of one partition is adjacent to every vertex of the other. A complete bipartite graph with bipartition (X, Y ) is denoted by Ka,b , where a = |X|, b = |Y |.

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A bipartite graph G is said to be a complete bipartite graph if every vertex of one partition is adjacent to every vertex of the other. A complete bipartite graph with bipartition (X, Y ) is denoted by Ka,b , where a = |X|, b = |Y |.

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An isomorphism of two graphs G and H is a bijective function f : V (G) → V (H) such that any two vertices u and v of G are adjacent in G if and only if f (u) and f (v) are adjacent in H.

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An isomorphism of two graphs G and H is a bijective function f : V (G) → V (H) such that any two vertices u and v of G are adjacent in G if and only if f (u) and f (v) are adjacent in H. That is, two graphs G and H are said to be isomorphic if (i) |V (G)| = |V (H)|, (ii) |E(G)| = |E(H)|, (iii) vi vj ∈ E(G) =⇒ f (vi )f (vj ) ∈ E(H). This bijection is commonly described as edge-preserving bijection.

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An isomorphism of two graphs G and H is a bijective function f : V (G) → V (H) such that any two vertices u and v of G are adjacent in G if and only if f (u) and f (v) are adjacent in H. That is, two graphs G and H are said to be isomorphic if (i) |V (G)| = |V (H)|, (ii) |E(G)| = |E(H)|, (iii) vi vj ∈ E(G) =⇒ f (vi )f (vj ) ∈ E(H). This bijection is commonly described as edge-preserving bijection. If an isomorphism exists between two graphs, then the graphs are called isomorphic graphs and denoted as G ' H. December 1, 2017

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¯ The complement or inverse of a graph G, denoted by G ¯ is a graph with V (G) = V (G) such that two distinct ver¯ are adjacent if and only if they are not adjacent tices of G in G.

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¯ The complement or inverse of a graph G, denoted by G ¯ is a graph with V (G) = V (G) such that two distinct ver¯ are adjacent if and only if they are not adjacent tices of G in G.

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¯ The complement or inverse of a graph G, denoted by G ¯ such that two distinct veris a graph with V (G) = V (G) ¯ are adjacent if and only if they are not adjacent tices of G in G.

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¯ The complement or inverse of a graph G, denoted by G ¯ such that two distinct veris a graph with V (G) = V (G) ¯ are adjacent if and only if they are not adjacent tices of G in G. ¯ we have Note that for a graph G and its complement G, ¯ = Kn ; (i) G1 ∪ G ¯ (ii) V (G) = V (G); ¯ = E(Kn ); (iii) E(G) ∪ E(G) ¯ = |E(Kn )| = n . (iv) |E(G)| + |E(G)| 2

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v1

v1

v5

v2

v5

v2

v4

v3

v4

v3 ¯ (b) G

(a) G

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A graph G is said to be self-complementary if G is isomorphic to its complement.

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A graph G is said to be self-complementary if G is isomorphic to its complement. Note: For self complementary graphs of order n, |E(G)| = ¯ = n(n−1) . This implies, 4 divides either n or n−1. |E(G)| 4 That is, we have n ≡ 0, 1(mod 4).

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A graph G is said to be self-complementary if G is isomorphic to its complement. Note: For self complementary graphs of order n, |E(G)| = ¯ = n(n−1) . This implies, 4 divides either n or n−1. |E(G)| 4 That is, we have n ≡ 0, 1(mod 4).

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Illustration to Self-Complementary Graphs

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A walk in a graph G is an alternating sequence of vertices and connecting edges in G. In other words, a walk is any route through a graph from vertex to vertex along edges. If the starting and end vertices of a walk is the same, then such a trail is called a closed walk.

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A walk in a graph G is an alternating sequence of vertices and connecting edges in G. In other words, a walk is any route through a graph from vertex to vertex along edges. If the starting and end vertices of a walk is the same, then such a trail is called a closed walk. A trail is a walk that does not pass over the same edge twice. A trail might visit the same vertex twice, but only if it comes and goes from a different edge each time. A closed trail or a tour is a trail that begins and ends on the same vertex.

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A walk in a graph G is an alternating sequence of vertices and connecting edges in G. In other words, a walk is any route through a graph from vertex to vertex along edges. If the starting and end vertices of a walk is the same, then such a trail is called a closed walk. A trail is a walk that does not pass over the same edge twice. A trail might visit the same vertex twice, but only if it comes and goes from a different edge each time. A closed trail or a tour is a trail that begins and ends on the same vertex. A path is a walk that does not include any vertex twice, except that its first vertex might be the same as its last.A cycle or a circuit is a path that begins and ends on the same vertex. December 1, 2017

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The length of a walk or trail or path or cycle is the number of edges in it.

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The length of a walk or trail or path or cycle is the number of edges in it. A path of order n is denoted by Pn and a cycle of order n is denoted by Cn .

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The length of a walk or trail or path or cycle is the number of edges in it. A path of order n is denoted by Pn and a cycle of order n is denoted by Cn . Every edge of G can be considered as a path of length 1.

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The length of a walk or trail or path or cycle is the number of edges in it. A path of order n is denoted by Pn and a cycle of order n is denoted by Cn . Every edge of G can be considered as a path of length 1. Note that the length of a path on n vertices has length n − 1 and the length of a cycle of order n is n.

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The length of a walk or trail or path or cycle is the number of edges in it. A path of order n is denoted by Pn and a cycle of order n is denoted by Cn . Every edge of G can be considered as a path of length 1. Note that the length of a path on n vertices has length n − 1 and the length of a cycle of order n is n. The distance between two vertices u and v in a graph G, denoted by dG (u, v) or simply d(u, v), is the length (number of edges) of a shortest path (also called a graph geodesic) connecting them.

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Two vertices u and v are said to be connected if there exists a path between them.

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Two vertices u and v are said to be connected if there exists a path between them. If there is a path between two vertices u and v, then u is said to be reachable from v and vice versa.

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Two vertices u and v are said to be connected if there exists a path between them. If there is a path between two vertices u and v, then u is said to be reachable from v and vice versa. A graph G is said to be connected if there exists paths between any two vertices in G. A graph which is not connected is called a disconnected graph.

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Two vertices u and v are said to be connected if there exists a path between them. If there is a path between two vertices u and v, then u is said to be reachable from v and vice versa. A graph G is said to be connected if there exists paths between any two vertices in G. A graph which is not connected is called a disconnected graph. A component of a graph G is a maximal connected subgraph of G. A connected graph has only one component. A graph having more than one component is a disconnected graph. Each vertex belongs to exactly one connected component, so does each edge. December 1, 2017

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An edge e of a graph G is said to be a cut-edge or a bridge of G if G − e is disconnected.

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An edge e of a graph G is a cut-edge of G if and only if it is not contained in any cycle of G. December 1, 2017

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A subset F of the edge set E of a graph G is said to be an edge-cut of G if G − E is disconnected.

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A subset F of the edge set E of a graph G is said to be an edge-cut of G if G − E is disconnected.

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A vertex v of a graph G is said to be a cut-vertex of G if G − v is disconnected.

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A vertex v of a graph G is said to be a cut-vertex of G if G − v is disconnected.

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¨ The city of Konigsberg in Prussia (now Kaliningrad, Russia) was situated on the either sides of the Pregel River and included two large islands which were connected to each other and the mainlands by seven bridges (see the below picture).

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¨ Konigsberg Seven Bridege Problem The problem was to devise a walk through the city that would cross each bridge once and only once, subject to the following conditions:

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¨ Konigsberg Seven Bridege Problem The problem was to devise a walk through the city that would cross each bridge once and only once, subject to the following conditions: I

The islands could only be reached by the bridges; and

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¨ Konigsberg Seven Bridege Problem The problem was to devise a walk through the city that would cross each bridge once and only once, subject to the following conditions: I I

The islands could only be reached by the bridges; and Every bridge once accessed must be crossed to its other end.

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I

The islands could only be reached by the bridges; and Every bridge once accessed must be crossed to its other end. The starting and ending points of the walk shall be the same.

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I

The islands could only be reached by the bridges; and Every bridge once accessed must be crossed to its other end. The starting and ending points of the walk shall be the same.

¨ The Konigsberg seven bridge problem was instrumental to the origination of Graph Theory as a branch of modern Mathematics. December 1, 2017

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In 1736, Swiss Mathematician Leonard Euler introduced a graphical model to this problem by representing each land area by a vertex and each bridge by an edge connecting corresponding vertices (see the following figure). B

A

D

C

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In 1736, Swiss Mathematician Leonard Euler introduced a graphical model to this problem by representing each land area by a vertex and each bridge by an edge connecting corresponding vertices (see the following figure). B

A

D

C Using this graphical model, Euler proved that no such a walk (trail) exists. December 1, 2017

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An Eulerian trail or Euler walk in an undirected graph is a walk that uses each edge exactly once. If an Euler trail exists in a given graph G, then G is called a traversable graph or a semi-Eulerian graph.

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An Eulerian trail or Euler walk in an undirected graph is a walk that uses each edge exactly once. If an Euler trail exists in a given graph G, then G is called a traversable graph or a semi-Eulerian graph. An Eulerian cycle or Eulerian circuit or Euler tour in an undirected graph is a cycle that uses each edge exactly once. A graph which contains an Euler cycle is called Eulerian graph or a unicursal graph.

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An Eulerian trail or Euler walk in an undirected graph is a walk that uses each edge exactly once. If an Euler trail exists in a given graph G, then G is called a traversable graph or a semi-Eulerian graph. An Eulerian cycle or Eulerian circuit or Euler tour in an undirected graph is a cycle that uses each edge exactly once. A graph which contains an Euler cycle is called Eulerian graph or a unicursal graph. Euler proved that a connected graph G is Eulerian if and only if every vertex G is even degree.

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A Hamiltonian path (or traceable path) is a path in an undirected (or directed graph) that visits each vertex exactly once.

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A Hamiltonian path (or traceable path) is a path in an undirected (or directed graph) that visits each vertex exactly once. A graph that contains a Hamiltonian path is called a traceable graph.

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A Hamiltonian path (or traceable path) is a path in an undirected (or directed graph) that visits each vertex exactly once. A graph that contains a Hamiltonian path is called a traceable graph. A Hamiltonian cycle, or a Hamiltonian circuit, or a vertex tour or a graph cycle is a cycle that visits each vertex exactly once (except for the vertex that is both the start and end, which is visited twice).

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A Hamiltonian path (or traceable path) is a path in an undirected (or directed graph) that visits each vertex exactly once. A graph that contains a Hamiltonian path is called a traceable graph. A Hamiltonian cycle, or a Hamiltonian circuit, or a vertex tour or a graph cycle is a cycle that visits each vertex exactly once (except for the vertex that is both the start and end, which is visited twice). A graph that contains a Hamiltonian cycle is called a Hamiltonian graph.

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A Hamiltonian path (or traceable path) is a path in an undirected (or directed graph) that visits each vertex exactly once. A graph that contains a Hamiltonian path is called a traceable graph. A Hamiltonian cycle, or a Hamiltonian circuit, or a vertex tour or a graph cycle is a cycle that visits each vertex exactly once (except for the vertex that is both the start and end, which is visited twice). A graph that contains a Hamiltonian cycle is called a Hamiltonian graph. Hamiltonian graphs are named after the famous mathematician William Rowan Hamilton who invented the Hamilton’s puzzle, which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron. December 1, 2017

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Figure: Aa Hamilton cycle in a Dodecahedron

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Figure: A graph that is Eulerian and Hamiltonian

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Figure: A graph that is Eulerian and Hamiltonian

Figure: A graph that is Eulerian, but not Hamiltonian

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Figure: A graph that is Hamiltonian, but not Eulerian

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Figure: A graph that is Hamiltonian, but not Eulerian

Figure: A graph is neither Eulerian nor Hamiltonian

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A graph G is called a tree if it is connected and has no cycles. That is, a tree is a conncted acyclic (circuitless) graph.

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Figure: Examples to trees

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Figure: Examples to trees

An acyclic graph may possibly be a disconnected graph whose components are trees. Such graphs are called forests. December 1, 2017

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I

In a tree, there is exactly one path between every pair of its vertices.

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I

I

In a tree, there is exactly one path between every pair of its vertices. A tree with n vertices has n − 1 edges.

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I

I I

In a tree, there is exactly one path between every pair of its vertices. A tree with n vertices has n − 1 edges. Any connected graph with n vertices and n − 1 edges is a tree.

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I

I I

I

In a tree, there is exactly one path between every pair of its vertices. A tree with n vertices has n − 1 edges. Any connected graph with n vertices and n − 1 edges is a tree. A tree is minimally connected. That is, every edge of G is a cut edge.

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I

I I

I

I

In a tree, there is exactly one path between every pair of its vertices. A tree with n vertices has n − 1 edges. Any connected graph with n vertices and n − 1 edges is a tree. A tree is minimally connected. That is, every edge of G is a cut edge. Any tree with at least two vertices has at least two pendant vertices.

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A spanning tree of a connected graph G is a tree containing all the vertices of G.

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A spanning tree of a connected graph G is a tree containing all the vertices of G. Every connected graph G has a spanning tree.

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A spanning tree of a connected graph G is a tree containing all the vertices of G. Every connected graph G has a spanning tree. The spanning tree of a graph is the maximal tree subgraph of that graph. A spanning tree of a graph G is sometimes called the skeleton or the scaffold graph.

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A spanning tree of a connected graph G is a tree containing all the vertices of G. Every connected graph G has a spanning tree. The spanning tree of a graph is the maximal tree subgraph of that graph. A spanning tree of a graph G is sometimes called the skeleton or the scaffold graph. Let T be a spanning tree of a given graph G. Then, every edge of T is called a branch of T . An edge of G that is not in a spanning tree of G is called a chord (or a tie or a link).

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Every graph with n vertices and edges has n−1 branches and − n + 1 chords.

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Every graph with n vertices and edges has n−1 branches and − n + 1 chords. The number of branches in a spanning tree T of a graph G is called its rank, denoted by rank(G). The number of chords of a graph G is called its nullity, denoted by nullity(G).

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Every graph with n vertices and edges has n−1 branches and − n + 1 chords. The number of branches in a spanning tree T of a graph G is called its rank, denoted by rank(G). The number of chords of a graph G is called its nullity, denoted by nullity(G). Therefore, we have rank(G) + nullity(G) = |E(G)|, the size ofG .

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A rooted tree is a tree T in which one vertex is distinguished from all other vertices. This particular vertex is called the root of T . v5 v4 v5 v3 v4 v2 v2 v3 v1

v1 v4

v5

v2

v3 v1

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A binary tree is a rooted tree in which there is only one vertex of degree 2 and all other vertices have degree 3 or 1. The vertex having degree 2 serves as the root of a binary tree.

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A binary tree is a rooted tree in which there is only one vertex of degree 2 and all other vertices have degree 3 or 1. The vertex having degree 2 serves as the root of a binary tree. The number of vertices in a binary tree is odd.

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A binary tree is a rooted tree in which there is only one vertex of degree 2 and all other vertices have degree 3 or 1. The vertex having degree 2 serves as the root of a binary tree. The number of vertices in a binary tree is odd. A binary tree on n vertices has

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n+1 2

pendant vertices.


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n+1 2

pendant vertices.

A vertex v of a binary tree is said to be at level ` if its distance from the root is `.

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n+1 2

pendant vertices.

A vertex v of a binary tree is said to be at level ` if its distance from the root is `. An `-level binary tree is said to be a complete binary tree if it has 2k vertices at the k-th level for all 1 ≤ k ≤ `.

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level 4 level 3 level 2 level 1 level 0 Figure: A 9-vertex binary tree with level 4.

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A connected graph G is a tree if and only if adding an edge between any two vertices in G creates exactly one cycle (circuit).

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A connected graph G is a tree if and only if adding an edge between any two vertices in G creates exactly one cycle (circuit). Let G be a connected graph and T be a spanning tree of G. Then, adding a chord to T creates a unique cycle and in G. Such a chord is called a fundamental circuit or fundamental cycle in G.

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A connected graph G is a tree if and only if adding an edge between any two vertices in G creates exactly one cycle (circuit). Let G be a connected graph and T be a spanning tree of G. Then, adding a chord to T creates a unique cycle and in G. Such a chord is called a fundamental circuit or fundamental cycle in G. Any connected graph G with n vertices and edges components has − n + 1 fundamental cycles.

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A connected graph G is a tree if and only if adding an edge between any two vertices in G creates exactly one cycle (circuit). Let G be a connected graph and T be a spanning tree of G. Then, adding a chord to T creates a unique cycle and in G. Such a chord is called a fundamental circuit or fundamental cycle in G. Any connected graph G with n vertices and edges components has − n + 1 fundamental cycles. Any graph G with n vertices, edges and k components has − n + k fundamental cycles.

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Cut-sets Recall that an edge e of a graph G is a cut-edge of G if G − e is disconnected.

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Cut-sets Recall that an edge e of a graph G is a cut-edge of G if G − e is disconnected. A subset F of the edge set E of a graph G is said to be an edge-cut of G if G − E is disconnected.

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Cut-sets Recall that an edge e of a graph G is a cut-edge of G if G − e is disconnected. A subset F of the edge set E of a graph G is said to be an edge-cut of G if G − E is disconnected. A cut-set is a minimal edge-cut of G. That is, a cut-set of a graph G is a set of edges F of G whose removal makes the graph disconnected, provided the removal of no proper subset of F makes G disconnected.

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Cut-sets Recall that an edge e of a graph G is a cut-edge of G if G − e is disconnected. A subset F of the edge set E of a graph G is said to be an edge-cut of G if G − E is disconnected. A cut-set is a minimal edge-cut of G. That is, a cut-set of a graph G is a set of edges F of G whose removal makes the graph disconnected, provided the removal of no proper subset of F makes G disconnected. A cut-set also called a minimal cut-set, a proper cut-set, a simple cut-set, or a cocycle. Note that the edge-cut F in the above example is a cut set-set of G. December 1, 2017

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I

Every cut-set in a graph G must contain at least one branch of every spanning tree of G.

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I

I

Every cut-set in a graph G must contain at least one branch of every spanning tree of G. Every cycle (circuit) in G has even number of edges in common with any cut-set of G.

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I

I

I

Every cut-set in a graph G must contain at least one branch of every spanning tree of G. Every cycle (circuit) in G has even number of edges in common with any cut-set of G. A connected graph G (or a connected component of a graph) is said to be a separable graph if it has a cut vertex.

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I

I

I

I

Every cut-set in a graph G must contain at least one branch of every spanning tree of G. Every cycle (circuit) in G has even number of edges in common with any cut-set of G. A connected graph G (or a connected component of a graph) is said to be a separable graph if it has a cut vertex. A connected graph or a connected component of a graph, which is not separable, is called non-separable graph.

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I

I

I

I

I

Every cut-set in a graph G must contain at least one branch of every spanning tree of G. Every cycle (circuit) in G has even number of edges in common with any cut-set of G. A connected graph G (or a connected component of a graph) is said to be a separable graph if it has a cut vertex. A connected graph or a connected component of a graph, which is not separable, is called non-separable graph. A non-separable subgraph of a separable graph G is called a block of G.

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v2

v1

v2

v5

v3

v7 v1

v6 (a) A separable graph

v5

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v6

(b) A non-separable graph

Figure: Illustration to separable and non-separable graphs

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v5

v2 v2 v1 (a) Block 1

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v3 v3 (b) Block 2

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Suppose there are three cottages (or houses)- H1 , H2 and H3 - on a plane and each needs to be connected to three utilities, say gas (G), water (W) and electricity E. Without using a third dimension or sending any of the connections through another house or cottage, is there a way to make all nine connections without any of the lines crossing each other?

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Suppose there are three cottages (or houses)- H1 , H2 and H3 - on a plane and each needs to be connected to three utilities, say gas (G), water (W) and electricity E. Without using a third dimension or sending any of the connections through another house or cottage, is there a way to make all nine connections without any of the lines crossing each other? The problem can be modelled graphically as follows:

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H1

H2

H3

G

E

W

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The above graph is often referred to as the utility graph in reference to the problem.

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The above graph is often referred to as the utility graph in reference to the problem. In more formal graph-theoretic terms, the problem asks whether the complete bipartite graph K3,3 can be redrawn in such a way that no two edges of K3,3 crosses each other.

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The above graph is often referred to as the utility graph in reference to the problem. In more formal graph-theoretic terms, the problem asks whether the complete bipartite graph K3,3 can be redrawn in such a way that no two edges of K3,3 crosses each other. The three utilities problem was the motivation for the introduction of the concepts of planarity of graphs.

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A face of a graph G is the region formed by a cycle or parallel edges or loops in G. A face of a graph G is also called a window, a region or a mesh.

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A face of a graph G is the region formed by a cycle or parallel edges or loops in G. A face of a graph G is also called a window, a region or a mesh. A Jordan curve is a non-self-intersecting continuous closed curve in the plane. Cycles in a graph can be considered to be Jordan Curves.

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A face of a graph G is the region formed by a cycle or parallel edges or loops in G. A face of a graph G is also called a window, a region or a mesh. A Jordan curve is a non-self-intersecting continuous closed curve in the plane. Cycles in a graph can be considered to be Jordan Curves. Every Jordan curve J divides the plane into an interior region, denoted by IntJ , bounded by the curve and an exterior region, denoted by ExtJ containing all of the nearby and far away exterior points.

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A graph G is called a planar graph if it can be re-drawn on a plane without any crossovers (That is, in such a way that two edges intersect only at their end points). Such a representation is sometimes called a topological planar graph.

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A graph G is called a planar graph if it can be re-drawn on a plane without any crossovers (That is, in such a way that two edges intersect only at their end points). Such a representation is sometimes called a topological planar graph. Such a drawing of G, if exists, is called a plane graph or planar embedding of G.

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A graph G is called a planar graph if it can be re-drawn on a plane without any crossovers (That is, in such a way that two edges intersect only at their end points). Such a representation is sometimes called a topological planar graph. Such a drawing of G, if exists, is called a plane graph or planar embedding of G. The portion of the plane lying outside a graph embedded in a plane is infinite in its extent such a region is called an infinite region, outer region, exterior region or unbounded region.

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An embedding of a graph G on a surface S is a geometric representation of G drawn on the surface such that the curves representing any two edges of G do not intersect except at a point representing a vertex of G.

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An embedding of a graph G on a surface S is a geometric representation of G drawn on the surface such that the curves representing any two edges of G do not intersect except at a point representing a vertex of G. In order that a graph G is planar, we have to show that there exists a graph isomorphic to G that is embedded in a plane.

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An embedding of a graph G on a surface S is a geometric representation of G drawn on the surface such that the curves representing any two edges of G do not intersect except at a point representing a vertex of G. In order that a graph G is planar, we have to show that there exists a graph isomorphic to G that is embedded in a plane. Equivalently, a geometric graph G is nonplanar if none of the possible geometric representations of G can be embedded in a plane.

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The graphs depicted in Figure 16 are some examples of planar graphs (or plane graphs).

(a)

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(b)

(a)

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(b)

(a)

(c)

Figure: Illustration to planar (plane) graphs

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A graph can be embedded on the surface of a sphere if and only if it can be embedded in a plane.

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A graph can be embedded on the surface of a sphere if and only if it can be embedded in a plane. N

S ˆ P

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Sudev Fundamentals of Graph Figure: SphericalN.K. embedding of a point inTheory a plane.

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Jordan Curve Theorem: The Jordan curve theorem asserts that any path connecting a point of interior region of a Jordan curve J to a point exterior region of J intersects with the curve J somewhere.

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Jordan Curve Theorem: The Jordan curve theorem asserts that any path connecting a point of interior region of a Jordan curve J to a point exterior region of J intersects with the curve J somewhere. The complete graph K5 and the complete bipartite graph K3,3 are called Kuratowski’s graphs, after the Polish mathematician Kasimir Kuratowski, who found that these two graphs are nonplanar. The complete graph K5 and the complete bipartite graph K3,3 are nonplanar.

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v5 v4 v1 v3 v2

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v5 v4 v1 v3 v2

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v5 v4 v1 v3 v2

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v5 v4 v1 v3 v2

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The complete bipartite graph K3,3 is nonplanar.

Figure

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We observe that the two graphs K5 and K3,3 have the following common properties: (i) Both K5 and K3,3 are regular.

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We observe that the two graphs K5 and K3,3 have the following common properties: (i) Both K5 and K3,3 are regular. (ii) K5 and K3,3 are nonplanar.

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We observe that the two graphs K5 and K3,3 have the following common properties: (i) Both K5 and K3,3 are regular. (ii) K5 and K3,3 are nonplanar. (ii) Removal of one edge or a vertex makes each of K5 and K3,3 a planar graph.

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We observe that the two graphs K5 and K3,3 have the following common properties: (i) Both K5 and K3,3 are regular. (ii) K5 and K3,3 are nonplanar. (ii) Removal of one edge or a vertex makes each of K5 and K3,3 a planar graph. (iv) K5 is a nonplanar graph with the smallest number of vertices, and K3,3 is the nonplanar graph with smallest number of edges.

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I

Euler Theorem on Plane Graphs: If G is a connected plane graph, then |V (G)|+|F (G)|−|E(G)| = 2, where V, E, F are respectively the vertex set, edge set and set of faces of G.

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I

Euler Theorem on Plane Graphs: If G is a connected plane graph, then |V (G)|+|F (G)|−|E(G)| = 2, where V, E, F are respectively the vertex set, edge set and set of faces of G. If G is a planar graph without parallel edges on n vertices and edges, where ≥ 3, then ≤ 3n − 6.

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I

Euler Theorem on Plane Graphs: If G is a connected plane graph, then |V (G)|+|F (G)|−|E(G)| = 2, where V, E, F are respectively the vertex set, edge set and set of faces of G. If G is a planar graph without parallel edges on n vertices and edges, where ≥ 3, then ≤ 3n − 6.

I

If G is bipartite, then ≤ 2n − 4.

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Let G be a separable graph with blocks G1 , G2 , G3 , . . . , Gk . Then, we need to check the planarity of each block separately.

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Let G be a separable graph with blocks G1 , G2 , G3 , . . . , Gk . Then, we need to check the planarity of each block separately. S1: Note that addition or removal of self-loops do not affect planarity. So, if G has self-loops, remove all of them.

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Let G be a separable graph with blocks G1 , G2 , G3 , . . . , Gk . Then, we need to check the planarity of each block separately. S1: Note that addition or removal of self-loops do not affect planarity. So, if G has self-loops, remove all of them. S2: Similarly, parallel edges do not affect planarity. So, if G has parallel edges, remove all of them, keeping one edge between every pair of vertices.

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Let G be a separable graph with blocks G1 , G2 , G3 , . . . , Gk . Then, we need to check the planarity of each block separately. S1: Note that addition or removal of self-loops do not affect planarity. So, if G has self-loops, remove all of them. S2: Similarly, parallel edges do not affect planarity. So, if G has parallel edges, remove all of them, keeping one edge between every pair of vertices. S3: We observe that removal of vertices having degree 2 by merging the two edges incident on it, perform this action as far as possible.

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Repeated performance of these steps will reduce the order and size of the graph without affecting its planarity (see Figure below). After repeated application of elementary reduction, the given graph will be reduced to any one of the following cases,

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Repeated performance of these steps will reduce the order and size of the graph without affecting its planarity (see Figure below). After repeated application of elementary reduction, the given graph will be reduced to any one of the following cases, (i) A single edge K2 ; or

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Repeated performance of these steps will reduce the order and size of the graph without affecting its planarity (see Figure below). After repeated application of elementary reduction, the given graph will be reduced to any one of the following cases, (i) A single edge K2 ; or (ii) A complete graph K4 ; or

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Repeated performance of these steps will reduce the order and size of the graph without affecting its planarity (see Figure below). After repeated application of elementary reduction, the given graph will be reduced to any one of the following cases, (i) A single edge K2 ; or (ii) A complete graph K4 ; or (iii) A non-separable simple graph with n ≥ 5, ≥ 7.

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Repeated performance of these steps will reduce the order and size of the graph without affecting its planarity (see Figure below). After repeated application of elementary reduction, the given graph will be reduced to any one of the following cases, (i) A single edge K2 ; or (ii) A complete graph K4 ; or (iii) A non-separable simple graph with n ≥ 5, ≥ 7. If H is a graph obtained from a graph G by a series of elementary reductions, then G and H are said to be homeomorphic graphs. In this case, H is also called a topological minor of G.

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v4

v3

v1

v2 (a) G

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v4

v3

v1

v2 (a)

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v4

v3

v1

v2 (a)

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v1

v2 (b)

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v3

v1

v2 (a)

v3

v1

v3

v1 (b)

(c)

Figure: Step by step illustration of elemntary reduction of a graph.

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Kuratowski’s Theorem: A graph G planar if and only if it has no subdivisions of K5 and K3,3 .

(a)

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(a)

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(a)

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A graph colouring is an assignment of labels (Colours) to the elements (vertices, edges or faces) of a graph subject to certain constraints.

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A graph colouring is an assignment of labels (Colours) to the elements (vertices, edges or faces) of a graph subject to certain constraints. A vertex colouring is a way of Colouring the vertices of a graph such that no two adjacent vertices share the same Colour. That is, a vertex colouring can be considered as a function c : V (G) → C, where C = {c1 , c2 , . . . , cr } is a set of colours.

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A graph colouring is an assignment of labels (Colours) to the elements (vertices, edges or faces) of a graph subject to certain constraints. A vertex colouring is a way of Colouring the vertices of a graph such that no two adjacent vertices share the same Colour. That is, a vertex colouring can be considered as a function c : V (G) → C, where C = {c1 , c2 , . . . , cr } is a set of colours. Similarly, an edge colouring assigns a Colour to each edge so that no two adjacent edges share the same Colour, and a face colouring or map colouring of a planar graph assigns a Colour to each face or region so that no two faces that share a boundary have the same colour. December 1, 2017

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Graph Colouring

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A labeling scheme for vertex colouring of graphs is illustrated as follows: c1 c2 c3

c2 c2

c1 c3

c1

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By colouring, we mean vertex Colouring unless specified otherwise. If the graph G has a loop at the vertex v then v is adjacent to itself and hence no colouring of G is possible. To avoid this uninteresting case we will assume that in any vertex colouring context graphs have no loops.

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By colouring, we mean vertex Colouring unless specified otherwise. If the graph G has a loop at the vertex v then v is adjacent to itself and hence no colouring of G is possible. To avoid this uninteresting case we will assume that in any vertex colouring context graphs have no loops. A colouring using at most k Colours is called a (proper) k-colouring. The smallest number of Colours needed to Colour a graph G is called its chromatic number, and is often denoted χ(G) (read it as chi(G)).

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By colouring, we mean vertex Colouring unless specified otherwise. If the graph G has a loop at the vertex v then v is adjacent to itself and hence no colouring of G is possible. To avoid this uninteresting case we will assume that in any vertex colouring context graphs have no loops. A colouring using at most k Colours is called a (proper) k-colouring. The smallest number of Colours needed to Colour a graph G is called its chromatic number, and is often denoted χ(G) (read it as chi(G)). A graph that can be assigned a (proper) k-colouring is said to be a k-colourable graph. A graph is said to be k-chromatic if its chromatic number is exactly k.

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An independent set is a set of elements of G in which no two elements are adjacent.

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An independent set is a set of elements of G in which no two elements are adjacent. A subset of vertices of V (G) having the same Colour is called a Colour class. Every such class forms an independent set. Thus, a k-Colouring is the same as a partition of the vertex set into k independent sets.

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An independent set is a set of elements of G in which no two elements are adjacent. A subset of vertices of V (G) having the same Colour is called a Colour class. Every such class forms an independent set. Thus, a k-Colouring is the same as a partition of the vertex set into k independent sets. A graph G is called critical if for every proper subgraph H of G, χ(H) ≤ χ(G). Also, G is k-critical if it is kchromatic and critical.

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An independent set is a set of elements of G in which no two elements are adjacent. A subset of vertices of V (G) having the same Colour is called a Colour class. Every such class forms an independent set. Thus, a k-Colouring is the same as a partition of the vertex set into k independent sets. A graph G is called critical if for every proper subgraph H of G, χ(H) ≤ χ(G). Also, G is k-critical if it is kchromatic and critical. Assigning distinct Colours to distinct vertices always yields a proper Colouring and hence we have 1 ≤ χ(G) ≤ n.

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1. If the graph G has n vertices, χ(G) ≤ n.

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1. If the graph G has n vertices, χ(G) ≤ n. 2. If H is a subgraph of the graph G, then χ(H) ≤ χ(G).

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1. If the graph G has n vertices, χ(G) ≤ n. 2. If H is a subgraph of the graph G, then χ(H) ≤ χ(G). 3. χ(Kn ) = n for all n > 1.

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1. If the graph G has n vertices, χ(G) ≤ n. 2. If H is a subgraph of the graph G, then χ(H) ≤ χ(G). 3. χ(Kn ) = n for all n > 1. 4. If the graph G contains Kn as a subgraph, then χ(G) ≥ n.

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1. If the graph G has n vertices, χ(G) ≤ n. 2. If H is a subgraph of the graph G, then χ(H) ≤ χ(G). 3. χ(Kn ) = n for all n > 1. 4. If the graph G contains Kn as a subgraph, then χ(G) ≥ n. 5. If the graph G has G1 , G2 , G3 , . . . , Gn as its connected components, then χ(G) = max{χ(Gi ) : 1 ≤ i ≤ n}.

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Figure

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Let us now look at some simple examples.

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Let us now look at some simple examples. Firstly, if the graph G has no edges, then each vertex can be given the same colour, i.e., χ(G) = 1.

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Let us now look at some simple examples. Firstly, if the graph G has no edges, then each vertex can be given the same colour, i.e., χ(G) = 1. Now, let G = Cn , the cycle of length n, with vertex set {v1 , v2 , . . . , vn }. Assign colour 1 to v1 , then, v2 by colour 2, v3 by the colour 1 again and so on. We note that if n is odd, then vn needs a different colour 3. Thus, χ(Cn ) = 2 if n is even and χ(Cn ) = 3 if n is odd.

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Illustration

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A non-empty graph G is 2-colourable (that is, χ(G) = 2) if and only if it is bipartite.

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A non-empty graph G is 2-colourable (that is, χ(G) = 2) if and only if it is bipartite. Let G be a graph. Then χ(G) ≥ 3 if and only if G has an odd cycle.

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An edge colouring of a non-empty graph G assigns colours, usually denoted by 1, 2, 3, . . . (or c1 , c2 , c3 , . . .) to the edges of G, one colour per edge, so that adjacent edges are assigned different colours. A k-edge colouring of G is a colouring of G which consists of k different colours and in this case, G is said to be k-edge colourable. The edge chromatic number of a graph G,denoted by χ1 (G), is minimum number k for which there exists an kedge colouring of G. If χ1 (G) = k, we say that G is k-edge chromatic.

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Figure: Example to edge colouring

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Graph Colouring

Figure: Example to edge colouring (2)

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A map is defined to be a connected plane graph with no bridges (cut edges). A map G is said to be k-face colourable if we can colour its faces with at most k colours in such a way that no two adjacent faces (two faces sharing a common boundary edge) have the same colour. The Four Colour Conjecture: Every map is 4-face colourable. A map G is 2-face colourable if and only if it is an Eulerian graph.

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A map is defined to be a connected plane graph with no bridges (cut edges). A map G is said to be k-face colourable if we can colour its faces with at most k colours in such a way that no two adjacent faces (two faces sharing a common boundary edge) have the same colour. The Four Colour Conjecture: Every map is 4-face colourable. A map G is 2-face colourable if and only if it is an Eulerian graph. Every map G has a face colouring of at most six colours.

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Graph Colouring

Figure: Example to map colouring (1) December 1, 2017

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Graph Colouring

Figure: Example to map colouring (2)

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Fundamentals of Graph Theory

Fundamentals of Graph Theory

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