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A note on fragile graphs
Discrete Mathematics 249 (2002) 41–43
www.elsevier.com/locate/disc
A note on fragile graphs Guantao Chena; ∗;1 , Xingxing Yub; 2 a Deptartment
b School
of Mathematics and Statistics, Georgia State University, Atlanta, GA 30303, USA of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA
Received 27 August 1999; revised 12 December 2000; accepted 26 March 2001
Abstract In this note, we show that every graph on n vertices and at most 2n − 4 edges contains a vertex-cut S which is also an independent set of G. The result is best possible and answers a c 2002 Elsevier Science B.V. All rights reserved. question proposed by Caro in the a2rmative.
We will only consider 8nite graphs without loops or multiple edges. Given a graph G, the vertex set of G and the edge set of G will be denoted by V (G) and E(G), respectively. Let v(G) = |V (G)| and e(G) = |E(G)|. If G is connected, S ⊆ V (G) is a cut of G if G − S is no longer connected. A vertex set S ⊆ V (G) is independent if no two vertices of S are joined by an edge. A cut S of G which is also an independent set is called an independent cut. A simple connected graph on at least 3 vertices is called a fragile graph if it contains an independent cut. Caro and Yuster [3] pointed out that fragile graphs play a role in some decomposition algorithms. deFigueiredo and Klein [4] proved that deciding if a graph is fragile is NP-complete. Clearly, every triangle-free connected graph with at least 3 vertices is fragile. Brandstaedt et al. [1] proved that deciding if a K4 -free graph is fragile is also NP-complete. It is naturally expected that sparse graphs (graph with few edges) are fragile. However, the graph obtained from K2; n−2 by adding an edge between vertices of the part with just two vertices is not fragile. This graph has n vertices and 2n − 3 edges. On the other hand, the following beautiful conjecture is made by Caro. Conjecture 1 (Caro). Let G be a graph of n vertices. If e(G) 6 2n − 4, then G contains an independent cut.
This work was done during the 12th Cumberland Conference on Combinatorics, Graph Theory and Computing, May 20 –22, 1999, Louisville. ∗ Corresponding author. E-mail address: [email protected] (G. Chen). 1 Research partially supported under NSF Grant No. DMS-0070059. 2 Partially supported by NSF grant DMS 9970527. c 2002 Elsevier Science B.V. All rights reserved. 0012-365X/02/$ - see front matter PII: S 0 0 1 2 - 3 6 5 X ( 0 1 ) 0 0 2 2 6 - 6
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G. Chen, X. Yu / Discrete Mathematics 249 (2002) 41–43
The above example shows that the bound 2n − 4 is best possible if the conjecture is true. Caro [2] pointed out that every connected graph with n ¿ 3 vertices and at most 3(n − 1)=2 edges is fragile. The purpose of this note is to prove that Caro’s conjecture is true. To do so, we prove a stronger result for 2-connected graphs. Theorem 1. Let G be a 2-connected graph on n vertices. If e(G) 6 2n − 4, then for every vertex x ∈ V (G) there is an independent cut S of G with x ∈ S. Proof. We 8rst note that the only 2-connected graph of order n = 3 is a triangle, which contains 2n − 3 = 3 edges. The only 2-connected graph of order 4 with at most 4 = 2 × 4 − 4 edges is C4 , a cycle on 4 vertices. Theorem 1 is true for n = 4. Based on these two facts, we assume that n ¿ 5 and Theorem 1 is true for all 2-connected graphs of order less than n. Let G be a 2-connected graph on n ¿ 5 vertices with at most 2n − 4 edges and let x be a vertex of G. If x is not in a triangle, then N (x) is an independent cut of G not containing x and the result holds. Assume that xyz is a triangle of G. Let H = G=xy denote the graph obtained from G by contracting the edge xy to a single vertex x∗ and deleting multiple edges and loop. Since x and y have a common neighbor z, the following inequality holds: e(H ) 6 e(G) − 2 6 (2n − 4) − 2 = 2(n − 1) − 4: If H is 2-connected, H has an independent cut S not containing x∗ . Clearly, S is also an independent cut of G and x ∈ S. Thus, we may assume that H is not 2-connected and {x; y} is a cut of G. Since G is 2-connected, G contains two 2-connected induced subgraphs G1 and G2 such that G = G1 ∪ G2 and V (G1 ) ∩ V (G2 ) = {x; y} and |V (Gi )| ¿ 3 for each i = 1; 2. If G1 contains an independent cut S not containing x, then either x and y are in the same component of G1 −S or y is in S since xy ∈ G1 . In either case, S is an independent cut of G not containing x. So we may assume that every independent cut of G1 contains x. Thus, by the induction hypothesis, we have e(G1 ) ¿ 2|V (G1 )| − 3 (this condition is automatically satis8ed if |V (G1 )| = 3). Similarly, we have e(G2 ) ¿ 2|V (G2 )| − 3. Therefore, e(G) = e(G1 ) + e(G2 ) − 1 ¿ 2|V (G1 )| − 3 + 2|V (G2 )| − 3 − 1 = 2(|V (G1 )| + |V (G2 )|) − 7 = 2(n + 2) − 7 = 2n − 3; a contradiction. Corollary 2. Let G be a connected graph on n vertices. If e(G) 6 2n − 4, then G contains an independent cut.
G. Chen, X. Yu / Discrete Mathematics 249 (2002) 41–43
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Proof. The result is true if the connectivity of G is 1. If the connectivity of G is at least 2, the result follows directly from Theorem 1. Remark. A result obtained by G. Chen, R.J. Faudree, and M.S. Jacobson recently shows that no bound can be placed on the order of the independent cutset unless more severe restrictions on the size of the graph. References [1] A. Brandstaedt, F. Dragan, Van B. Le, T. Szyczack, On stable cutsets in graphs, preprint. University of Rostock, 1998. [2] Y. Caro, private communication. [3] Y. Caro, R. Yuster, Decomposition of slim graphs, Graphs Combin. 15 (1999) 5–19. [4] C. deFigueiredo, S. Klein, NP-completeness of multipartite cutset testing, Congr. Numer. 119 (1996) 217–222.
www.elsevier.com/locate/disc
A note on fragile graphs Guantao Chena; ∗;1 , Xingxing Yub; 2 a Deptartment
b School
of Mathematics and Statistics, Georgia State University, Atlanta, GA 30303, USA of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA
Received 27 August 1999; revised 12 December 2000; accepted 26 March 2001
Abstract In this note, we show that every graph on n vertices and at most 2n − 4 edges contains a vertex-cut S which is also an independent set of G. The result is best possible and answers a c 2002 Elsevier Science B.V. All rights reserved. question proposed by Caro in the a2rmative.
We will only consider 8nite graphs without loops or multiple edges. Given a graph G, the vertex set of G and the edge set of G will be denoted by V (G) and E(G), respectively. Let v(G) = |V (G)| and e(G) = |E(G)|. If G is connected, S ⊆ V (G) is a cut of G if G − S is no longer connected. A vertex set S ⊆ V (G) is independent if no two vertices of S are joined by an edge. A cut S of G which is also an independent set is called an independent cut. A simple connected graph on at least 3 vertices is called a fragile graph if it contains an independent cut. Caro and Yuster [3] pointed out that fragile graphs play a role in some decomposition algorithms. deFigueiredo and Klein [4] proved that deciding if a graph is fragile is NP-complete. Clearly, every triangle-free connected graph with at least 3 vertices is fragile. Brandstaedt et al. [1] proved that deciding if a K4 -free graph is fragile is also NP-complete. It is naturally expected that sparse graphs (graph with few edges) are fragile. However, the graph obtained from K2; n−2 by adding an edge between vertices of the part with just two vertices is not fragile. This graph has n vertices and 2n − 3 edges. On the other hand, the following beautiful conjecture is made by Caro. Conjecture 1 (Caro). Let G be a graph of n vertices. If e(G) 6 2n − 4, then G contains an independent cut.
This work was done during the 12th Cumberland Conference on Combinatorics, Graph Theory and Computing, May 20 –22, 1999, Louisville. ∗ Corresponding author. E-mail address: [email protected] (G. Chen). 1 Research partially supported under NSF Grant No. DMS-0070059. 2 Partially supported by NSF grant DMS 9970527. c 2002 Elsevier Science B.V. All rights reserved. 0012-365X/02/$ - see front matter PII: S 0 0 1 2 - 3 6 5 X ( 0 1 ) 0 0 2 2 6 - 6
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G. Chen, X. Yu / Discrete Mathematics 249 (2002) 41–43
The above example shows that the bound 2n − 4 is best possible if the conjecture is true. Caro [2] pointed out that every connected graph with n ¿ 3 vertices and at most 3(n − 1)=2 edges is fragile. The purpose of this note is to prove that Caro’s conjecture is true. To do so, we prove a stronger result for 2-connected graphs. Theorem 1. Let G be a 2-connected graph on n vertices. If e(G) 6 2n − 4, then for every vertex x ∈ V (G) there is an independent cut S of G with x ∈ S. Proof. We 8rst note that the only 2-connected graph of order n = 3 is a triangle, which contains 2n − 3 = 3 edges. The only 2-connected graph of order 4 with at most 4 = 2 × 4 − 4 edges is C4 , a cycle on 4 vertices. Theorem 1 is true for n = 4. Based on these two facts, we assume that n ¿ 5 and Theorem 1 is true for all 2-connected graphs of order less than n. Let G be a 2-connected graph on n ¿ 5 vertices with at most 2n − 4 edges and let x be a vertex of G. If x is not in a triangle, then N (x) is an independent cut of G not containing x and the result holds. Assume that xyz is a triangle of G. Let H = G=xy denote the graph obtained from G by contracting the edge xy to a single vertex x∗ and deleting multiple edges and loop. Since x and y have a common neighbor z, the following inequality holds: e(H ) 6 e(G) − 2 6 (2n − 4) − 2 = 2(n − 1) − 4: If H is 2-connected, H has an independent cut S not containing x∗ . Clearly, S is also an independent cut of G and x ∈ S. Thus, we may assume that H is not 2-connected and {x; y} is a cut of G. Since G is 2-connected, G contains two 2-connected induced subgraphs G1 and G2 such that G = G1 ∪ G2 and V (G1 ) ∩ V (G2 ) = {x; y} and |V (Gi )| ¿ 3 for each i = 1; 2. If G1 contains an independent cut S not containing x, then either x and y are in the same component of G1 −S or y is in S since xy ∈ G1 . In either case, S is an independent cut of G not containing x. So we may assume that every independent cut of G1 contains x. Thus, by the induction hypothesis, we have e(G1 ) ¿ 2|V (G1 )| − 3 (this condition is automatically satis8ed if |V (G1 )| = 3). Similarly, we have e(G2 ) ¿ 2|V (G2 )| − 3. Therefore, e(G) = e(G1 ) + e(G2 ) − 1 ¿ 2|V (G1 )| − 3 + 2|V (G2 )| − 3 − 1 = 2(|V (G1 )| + |V (G2 )|) − 7 = 2(n + 2) − 7 = 2n − 3; a contradiction. Corollary 2. Let G be a connected graph on n vertices. If e(G) 6 2n − 4, then G contains an independent cut.
G. Chen, X. Yu / Discrete Mathematics 249 (2002) 41–43
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Proof. The result is true if the connectivity of G is 1. If the connectivity of G is at least 2, the result follows directly from Theorem 1. Remark. A result obtained by G. Chen, R.J. Faudree, and M.S. Jacobson recently shows that no bound can be placed on the order of the independent cutset unless more severe restrictions on the size of the graph. References [1] A. Brandstaedt, F. Dragan, Van B. Le, T. Szyczack, On stable cutsets in graphs, preprint. University of Rostock, 1998. [2] Y. Caro, private communication. [3] Y. Caro, R. Yuster, Decomposition of slim graphs, Graphs Combin. 15 (1999) 5–19. [4] C. deFigueiredo, S. Klein, NP-completeness of multipartite cutset testing, Congr. Numer. 119 (1996) 217–222.