About presentation of a digraph by dim 2 poset Anatoly D. Plotnikov Vinnitsa Financial-Economic University, 71-A, st. Pirogova,Vinnitsa, 21037, UKRAINE
[email protected] May, 2005
Abstract
A binary relation R on a set A is any subset of A A. If the relation R is antisymmetric and transitive then R is a partially ordered set (poset). If (a b) 2 R the we write aRb or a R b. We also use a notation R = (A R ). A poset L = (A L ) is a chain if we have either a L b or b L a for any a b 2 A. A poset R has the dimension dim R = 2 if there exist two chains L1 and L2 such that R = L1 \ L2 3]. A digraph G is called orderable (oDAG) if there exists are dim 2 poset such that its Hasse diagram coincide with the digraph G. In this paper, we determine conditions when a digraph G may be presented by the corresponding dim 2 poset R.
MSC 2000: 05C20, 68R10, 94C15 KEYWORDS: binary relation, orderable graph, dimension poset, Hasse
diagram.
1 Basic denitions We shall remind some the well-known notions 3, 4]. A directed graph or digraph G~ is a pair (V E~ ) consisting of a vertex set V , and a set E~ of ordered pairs of vertices from A called by (ordered) edges. The rst vertex of the ordered pair is the tail of the edge, and the second is the head. Both these vertices are called the endpoints. 1
3cc
4 c 2
~
1 (a)G
3cc
TT c 4 TT 2 G~ t T 1
(b) Fig. 1: Digraph G~ and its transitive closure graph G~ t Edges from E~ we shall also call arcs. In a digraph, a loop is an edge whose endpoints are equal. Multiple edges are edges having the same ordered pair of endpoints. A digraph is simple if each ordered pair is the head and tail of at most one edge one loop may be present at each vertex. In the simple digraph, we write (u v ) for an edge with tail u and head v. If there is an edge from u to v, then v is a successor of u, and u is a predecessor of v . We write also u ! v if there is an edge from u to v . A digraph is a path if it is a simple digraph whose vertices can be linearly ordered so that there is an edge with tail u and head v if and only if v immediately follows u in the vertex ordering. A cycle is dened similarly using an ordering of the vertices on a circle. Path of G~ from the vertex x to vertex y we denote by p(x y ). A graph with no cycle is acyclic. A forest is an acyclic graph. A tree is connected acyclic graph. A digraph G~ is called transitive if for each arcs u ! v and v ! w from E~ there exists arc u ! w in E~ . We denote by Dn the set of all acyclic directed n-vertex graphs without loops and multiple edges. In general case, a digraph G~ 2 Dn is not transitive. For any path p(x y ) ~ of G, we shall introduce the arc (x y ) if such arc does not exist in G~ . As a result, we shall obtain a graph G~ t = (V E~ t), which is called a transitive closure digraph. Any arc of the graph G~ t, that does not exist in the digraph G~ , is called ctitious. For example, Fig. 1 (a) shows the digraph G~ , and Fig.1 (b) shows its transitive closure graph G~ t (arcs of G~ and G~ t are oriented bottom-up). Here, arcs (1 3), (1 4) are ctitious arc of G~ t . Let A be a some set. Any subset R A A is called a binary relation on the set A 2]. If (a b) 2 R then we write aRb, where a b 2 A. If a is not related to b by R, we write aRb. A relation R on a set A is reexive if aRb for every a 2 A. A relation R 2
on a set A is antisymmetric if whenever aRb and bRa then a = b. At last, a relation R on a set A is transitive if whenever aRb and bRc then aRc. A relation R on a set A is called a partial order if R is reexive, antisymmetric, and transitive. The set A together with the partial order R is called a partially ordered set, or a poset, and we will be denote this poset by (A R) or (A R). If (a b) 2 R, where R is a poset, then we also write a R b. Two elements a b 2 A such that a R b in R or b R b in R are said to be comparable otherwise, they are said to be incomparable. If P and Q are partial orders on the same set A, Q is said to be an extension of P if a P b implies a Q b, for all a b 2 A. A poset L is a chain, or a linear order if we have either a L b or b L a for any a b 2 A. If Q is a linear order then it is a linear extension of P . The dimension dim R of R is the least positive integer s for which there exists Ta family F = (L1 L2 : : : Ls) of linear extensions of R such that R = si=1 Li 3]. A family F = (L1 L2 : : : Ls) of linear orders on A is called a realizer of R on A if
R=
\s L :
i=1
i
Let R is a partial order on a set A. This order may be represented by a graph which is called Hasse diagram. Each element of A is presented by a vertex of the Hasse diagram. For each a b 2 A, if a R b then the vertex a is drawn lower then the vertex b. The vertices a, b are joined by an edge if and only if there does not exist a vertex c such that a c, and c b. Hasse diagrams are also called upward drawings. For example, let we have the order X = f1 2 3 4g R = f(1 2) (1 3) (1 4) (2 3) (2 4)g Then we have the Hasse diagram
3cc
4 c2 1
Note, the pairs (1 3), (1 4) are not mapped here. 3
2 Statement of the problem Let there be a digraph G~ = (V E~ ) 2 Dn without loops and multiple edges. Each digraph G~ 2 Dn will be called DAG. A digraph G~ 2 Dn will be called orderable (oDAG) if there exists are dim 2 poset such that its Hasse diagram coincide with the digraph G~ . For the given digraph G~ 2 Dn , it is required to determine conditions when it may be represented by the corresponding dim 2 poset (see, for example, 1]).
3 oDAG existence problem In this section, we shall determine a class of DAGs, which may be represented as oDAGs. To construct a dim 2 poset, corresponding to the given digraph G~ 2 Dn , we should have two some linear orders L1 = (X
