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ADDING MINIMUM NUMBER OF EDGES TO INCREASE THE BANDWIDTH OF GRAPHS Yung-Ling Lai, Chang-Sin Tian, Ting-Chun Ko Department of Computer Science and Information Engineering, National Chiayi University, Chiayi, Taiwan {yllai, s0930309,
[email protected]} Abstract: It is known that the bandwidth decision problem for an arbitrary graph is NPcomplete [11]. In fact, the problem is NP-complete even for trees of maximum degree 3 [8]. Local operations on a graph are transformations from a graph to another graph by adding or subtracting an edge or a vertex. Under a local operation, the bandwidth of a graph may increase, decrease or stay unchanged. It is trivial that adding edges to a graph won’t decrease the bandwidth of the graph. In 1986, Chvátalová and Opatrny [5] determined the maximum possible value of bandwidth increase as a function of B (G ) and the distance dG (u , v ) when adding edge uv to a graph G. In 1995, Wang, West, and Yao [12] determined the upper bound of bandwidth increasing under a single edge addition. In this paper, we define the edge addition number ad (G ) of G to be
ad (G ) = min { X : X ⊆ E (G ) and B (G + X ) > B (G )} and determine ad (G ) when G is a path, a cycle, or a complete bipartite graph, and we also generalize the result to complete r-partite graphs.
Introduction and Terminologies
For a graph G , V (G ) denotes the set of vertices of G and E (G ) denotes the set of edges of G . Let G be a graph on n vertices. A one-to-one mapping f : V (G ) → {1, 2,L , n} is called a proper numbering of G . The bandwidth B f (G ) of a proper numbering f of G is the number B f (G ) = max { f (u ) − f (v) : uv ∈ E (G )} and the bandwidth B (G ) of G is the number B (G ) = min { B f (G ) : f is a proper numbering of G} . A proper numbering f is called a bandwidth numbering of G , if B f (G ) = B(G ) .
The decision problem corresponding to find the bandwidth of an arbitrary graph was shown to be NP-Complete by Papadimitriou [11]. Garey et. al. [8] showed that the problem is NPComplete even for trees of maximum degree 3. The results and applications on bandwidth can be found in surveys [2, 3,10]. The local operations in a graph refer to the elementary refinement, contraction, addition of an edge and merger of two vertices. For a graph G , let e = uv ∉ E (G ) , we use G + e to denote the graph obtained from G by adding edge e, such that
V (G + e) = V (G ),
E (G + e) = E (G ) U {e} .
The graph obtained from G by adding all the edges in the set of X ⊆ E (G ) is denoted by G + X , where V (G + X ) = V (G ),
E (G + X ) = E (G ) U X .
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The relation between graph bandwidth and number of edges in the graph was first studied by Chvátalová. In 1980, Chvátalová [4] observed that adding an edge to a simple graph may increase the bandwidth by more than one. In 1986, Chvátalová and Opatrny [5] determined the maximum value of B (G + uv) as a function of B (G ) and the distance dG (u, v) . In 1989 and 1992, Dutton and Brigham [6] and Alavi et. al. [1] established some bounds for the size of graph with given bandwidth. In 1995, Wang, West, and Yao [12] determined maximum possible value of B(G + e) in terms of B (G ) and V (G ) . Looking at the problem in another angel, we are interested in how many edges have to be added into a graph to cause the bandwidth of the resulting graph be greater than the bandwidth of the original graph. For a graph G ≠ K n , let
{
the edge addition number ad (G ) of G to be defined as
}
ad (G ) = min X : X ⊆ E (G ) and B(G + X ) > B(G ) .
Then it is trivial that for all graph
G ≠ K n , ad (G ) ≥ 1 . In this paper, we determine ad (G ) when G is a path, a cycle, or a complete bipartite graph, and we also generalize the result of complete bipartite graphs to complete r-partite graphs. Paths and Cycles Theorem 1. Let G = Pn = v1v2 v3 L vn , where n ≥ 3 . Then ad (G ) = 1 . Proof: This is trivial since by adding B ( Pn + e) = B(Cn ) = 2 > 1 = B( Pn ) . Hence ad ( Pn ) = 1 . □
edge
e = v1vn
,
we
have
Theorem 2. Let Cn = v1v2 v3 L vn v1 be a cycle of n vertices, where n ≥ 4 . Then ad (Cn ) = 2 . Proof: First we show that adding any one edge to Cn is not possible to increase the bandwidth of the resulting graph. Define a proper numbering f of Cn as: 2i − 1, f (vi ) = 2(n − i + 1),
Then B f (Cn ) = B(Cn ) = 2 .
1 ≤ i ≤ n2 ; n2 < i ≤ n.
Let G = Cn + e where e = vs vt , s < t .
Consider a proper
numbering g of G, which is defined as follows: If f (vs ) − f (vt ) = 1 , then g (vi ) = f (vi ) for 1 ≤ i ≤ n . f (v ), If f (vs ) − f (vt ) > 1 , let r = s 2+t − 1 , then g (vi ) = i − r f (vn +i − r ),
i − r > 0, i − r ≤ 0.
Then we have Bg (G ) = B(Cn ) = 2 , which implies ad (Cn ) ≥ 2 . To see that ad (Cn ) ≤ 2 , for n = 4 , let X = E (C4 ) then X = 2 and C4 + X = K 4 . So we have B (C4 + X ) = 3 > 2 = B(C4 ) . For n ≥ 5 , let X ′ = {v2 vn −1 , v3vn } , then X ′ ⊆ E (Cn ), X ′ = 2 and B (Cn + X ′) = 3 > 2 = B(Cn ) , which implies ad (Cn ) ≤ 2 . That completes the proof. □
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Complete Bipartite Graphs
In this section, we present the edge addition number of complete bipartite graphs and generalize the results to complete r-partite graphs. For a bipartite graph G = K n ,m , let Vn = {v1 , v2 , v3 ,L , vn } be the partite set with n vertices and U m = {u1 , u2 , u3 ,L , um } be the
partite set with m vertices and the subgraph induced by S ⊆ V (G ) is denoted as < S > . Let G = K n1 ,n2 ,L,nr be a complete r-partite graph with
Proposition 1. (From [7])
n1 ≥ n2 ≥ L ≥ nr and r ≥ 2 . Then B(G ) = ∑ i =1 ni − n12+1 . r
Proposition 2. (From [6]) Let G be a graph with n vertex. Then E (G ) ≥ 2 B (G ) − 1 and
E (G ) = 2 B (G ) − 1 if and only if G is K1,2 B (G ) −1 or K 3 . Theorem 3. Let G = K n ,m be a complete bipartite graph with 1 ≤ m < n such that n is even.
Then ad (G ) = n − 1 . Proof: First
we
X ⊆ E (G ), X = n − 1
show
ad (G ) ≤ n − 1
that
G + X = K n −1, m,1
and
. .
Let By
X = {v1vi 2 ≤ i ≤ n} proposition
1
then we
have B (G + X ) = m + n − n −21+1 > m + n − n2+1 = B (G ) , which implies ad (G ) ≤ n − 1 . To see that ad (G ) ≥ n − 1 , suppose to the contrary that ad (G ) ≤ n − 2 , then there exists X ′ ⊆ E (G ) , X ′ = n − 2 such that B (G + X ′) > B(G ) . Define a proper numbering f of G as follows: i f (vi ) = i + m
for 1 ≤ i ≤ for
n 2
n 2
that will not increase the bandwidth of the resulting graph, so we may assume X ′ ⊆ E (< Vn >) . Since < Vn > = K n , according to the numbering f, this problem can be transform to the problem of finding minimum number of edges on n vertices to have bandwidth n2 . By proposition 2, we have X ′ ≥ n − 1 , which is a contradiction.
□
Since the bandwidth of a complete r-partite graph is only depended on the maximum Cardinality of the partite sets, corollary 1 comes directly from theorem 3. Corollary 1. Let G = K n1 ,n2 ,L,nr be a complete r-partite graph with n1 > n2 ≥ L ≥ nr such that
n1 is even. Then ad (G ) = n1 − 1 . Proposition 3. (from [1]) The minimum size required for a graph of order 5 and bandwidth 3 is 6; The minimum size required for a graph of order 7 and bandwidth 4 is 9. Lemma 1. Let G be a graph of order n such that n ≥ 9 and n is odd. Then the minimum size required for G to have bandwidth n2+1 is 2(n − 2) and the only extremal graph is K n − 2,2 .
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In 2000, Hao [9] provided the minimum size of a graph with order n and bandwidth n +2 3 and bandwidth n2 + 2 . Lemma 1 can be proved by a very similar method as described in the proof of theorem 1 in [9]. Theorem 4. Let G = K n ,m be a complete bipartite graph with 1 ≤ m < n − 1 such that n is
odd. 2(n − 2) Then ad (G ) = 9 Proof:
if n ≠ 7, . if n = 7.
For the case n ≠ 7 , let X = {vi v j :1 ≤ i ≤ 2, 3 ≤ j ≤ n} then X ⊆ E (G ), X = 2(n − 2)
and G + X = K n − 2,m ,2 . By proposition 1 we have B (G + X ) = m + n − n −22+1 = n + m − n2−1 > n + m − n2+1 = B (G ) , which implies ad (G ) ≤ 2(n − 2) . Let X ′ be the edge set with minimum cardinality such that B (G + X ′) > B(G ) . Same as in theorem 3, we may assume X ′ ⊆ E (< Vn >) and the problem can be transformed to finding the minimum size of graph
with order n and bandwidth
n +1 2
.
By proposition 3 and lemma 1, we have
ad (G ) = X ′ ≥ 2(n − 2) . Hence ad (G ) = 2(n − 2) for n ≠ 7 .
For
n=7
,
Consider
X = {vi v j :1 ≤ i ≤ 3, 4 ≤ j ≤ 6}
then
X ⊆ E (G ), X = 9
and
B (G + X ) = 4 + m > 3 + m = B(G ) , which implies ad (G ) ≤ 9 . Same as above, we may ssume X ′ ⊆ E (< V7 >) and the problem can be transformed to finding the minimum size of graph
with order 7 and bandwidth 4.
By proposition 3, we have ad (G ) = X ′ ≥ 9 .
ad (G ) = 9 for n = 7 , which completes the proof.
Hence □
As long as the maximum cardinality of the partite set is odd and is at least 2 more than the next highest of the partite set’s cardinality, corollary 2 can be generated by theorem 4. Corollary 2. Let G = K n1 ,n2 ,L,nr be a complete r-partite graph with n1 − 1 > n2 ≥ L ≥ nr such
that n1 is odd. Then ad (G ) = 2(n1 − 2) . Theorem 5. Let G = K n,n −1 be a complete bipartite graph such that n is odd. Then
3(n − 2) for n ≠ 7, ad (G ) = for n = 7. 14 Proof:
{
}
For n ≠ 7 , let X = vi v j 1 ≤ i ≤ 2, 3 ≤ j ≤ n ∪ {u1uk 2 ≤ k ≤ n − 1} then X ⊆ E (G ),
X = 3(n − 2) and G + X = K n − 2, n − 2,2,1 . By proposition 1 we have B (G + X ) = 2n − 1 − n2−1 > 2n − 1 − n2+1 = B (G ) , which implies ad (G ) ≤ 3(n − 2) .
For n = 7 , consider
X = {vi v j :1 ≤ i ≤ 3, 4 ≤ j ≤ 6} U {u1uk : 2 ≤ k ≤ 6} , then X ⊆ E (G ), X = 14 and B (G + X )
= 10 > 9 = B(G ) , which implies ad (G ) ≤ 14 . On the other hand, ad (G ) ≥ 3(n − 2) for n ≠ 7 and ad (G ) ≥ 14 for n = 7 can be proved by applying theorem 4 and corollary 1. So we 3(n − 2) for n ≠ 7, □ have ad (G ) = for n = 7. 14
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When the maximum cardinality of the partite set is odd and the next highest cardinality is only 1 less than the maximum, we will have to make the complete r-partite graph to complete (r+s+t)-partite graph where s and t are indicated in corollary 3. Corollary 3. Let G = K n1 , n2 ,L,nr be a complete r-partite graph with n1 ≥ n2 ≥ L ≥ nr such that
{
}
n1 is odd and n1 ≠ 7 . Let t = n j n j = n1 , and s = {ni ni = n1 − 1} . Then (2t + s )(n1 − 2) for n1 ≠ 7, ad (G ) = for n1 = 7. 9t + 5s
Theorem 6. Let G = K n ,n be a complete bipartite graph with equal partite sets. Then
4(n − 2) ad (G ) = 18 2(n − 1) Proof:
if n is odd and n ≠ 7, if n = 7, if n is even.
{
}
For even n, consider X = {v1vi 2 ≤ i ≤ n} ∪ u1u j 2 ≤ i ≤ n , then X ⊆ E (G ) ,
X = 2(n − 1) and G + X = K n −1,n −1,1,1 . > n + m − n2+1 = B(G ) ,
which
By proposition 1 we have B (G + X ) = m + n − n2
implies
ad (G ) ≤ 2(n − 1) .
Similar
to
theorem3,
ad (G ) ≥ 2(n − 1) can be shown by applying proposition 2 and considering the fact that B (G1 ∪ G2 ) = max { B (G1 ), B (G2 )} . For G = K 7,7 , consider X = {vi v j :1 ≤ i ≤ 3, 4 ≤ j ≤ 6} U {ui u j :1 ≤ i ≤ 3, 4 ≤ j ≤ 6} ,
X ⊆ E (G ), X = 18 and
then
B (G + X ) = 11 > 10 = B(G ) , which implies ad ( K 7,7 ) ≤ 18 . For odd n and n ≠ 7 , consider
{
} {
}
X = vi v j 1 ≤ i ≤ 2, 3 ≤ j ≤ n ∪ ui u j 1 ≤ i ≤ 2, 3 ≤ j ≤ n , then and
G + X = K n − 2,n − 2,2,2 .
By
proposition
1
we
X ⊆ E (G ),
have
X = 4(n − 2)
B (G + X ) = n + m − n2−1
> n + m − n2+1 = B(G ) , which implies ad (G ) ≤ 4(n − 2) for n ≠ 7 .
On the other hand,
ad (G ) ≥ 4(n − 2) for n ≠ 7 and ad ( K 7,7 ) ≥ 18 can be shown in a similar way as in theorem 4 with the fact that B (G1 ∪ G2 ) = max { B (G1 ), B (G2 )} .
□
For the general case in complete r-partite graph, if the maximum cardinality of the partite set is odd, the case has included in corollary 3. If the maximum cardinality of the partite set is even, it is given by corollary 4 which can be obtained by the similar way as in theorem 6. Corollary 4. Let
G = K n1 , n2 ,L,nr
be
a
complete
r-partite
graph
with
n1 = L = nk > nk +1 ≥ L ≥ nr for some k , 1 ≤ k ≤ r , such that n1 is even. ad (G ) = k (n1 − 1) .
Then
Conclusions
How does the number of edges affect the bandwidth in a graph is an interesting problem which gets researcher’s attention since 1980. In this paper, we define a new parameter named “edge addition number” related to this issue. Edge addition number indicates the minimum number of edges which have to be added into a graph to cause the bandwidth of the resulting graph be greater than the bandwidth in the original graph.
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For the simple graphs like paths and cycles, the edge addition numbers are 1 and 2 respectively. For the complete bipartite graph and the complete r-partite graph in general, the edge addition number is determined by the value of n1 which is the maximum cardinality of the partite set. We discussed different cases of n1 , when n1 is even or odd and when n1 is greater then the second highest cardinality of the partite set by 1 or more. References
[1]. Y. Alavi, J. Liu, and J. McCanna, On the minimum size of graphs with a given bandwidth, Bulletin of the ICA, Vol. 6 (1992), 22-32. [2]. P.Z. Chinn, J. Chvátalová, A.K. Dewdney, and N.E. Gibbs, The bandwidth problem for graphs and matrices—A survey, J. Graph Theory, Vol. 6 (1982), 223-254. [3]. F.R.K. Chung, Labeling of graphs, in Selected Topics in Graph Theory, Vol. 3, (L.W. Beineke and R.J. Wilson, eds.), (Academic Press, 1988), 151-168. [4]. J. Chvátalová, On the bandwidth problem for graphs, Ph.D. thesis, University of Waterloo, Canada (1980). [5]. J. Chvátalová, J. Opatrny, The bandwidth problem and operations on graphs, Discrete Math. Vol. 61 (1986), 141-150. [6]. R. D. Dutton, and R. C. Brigham, On the size of graphs of a given bandwidth, Discrete Mathematics, Vol. 76 (1989), 191-195. [7]. P. G. Eitner, The bandwidth of the complete multipartite graph. Presented at the Toledo Symposium on Applications of Graph Theory (1979). [8]. M. R. Garey, R. L. Graham, D. S. Johnson, and D. E. Knuth, “Complexity results for bandwidth minimization”, SIAM Journal on Applied Mathematics, vol. 34, pp. 477-495, 1978. [9]. J. Hao, “Two results on extremal bandwidth problem,” Mathematica Applicata, Vol. 13, no. 3, (2000), pp. 73-78. [10]. Y. L. Lai and K. Williams, “A survey of solved problems and applications on bandwidth, edgesum, and profile of graphs,” Journal of Graph Theory, Vol. 31, no. 2, pp. 75-94, 1999. [11]. C. H. Papadimitriou, “The NP-completeness of the bandwidth minimization problem,” Computing, Vol. 16, pp. 263-270, 1976. [12]. J.-F. Wang, D.B. West, B. Yao, Maximum bandwidth under edge addition, J. Graph Theory, Vol. 20 (1995), 87-90.
