COCOON 2004: Computing and Combinatorics pp 151-160

# On the Arrangement of Cliques in Chordal Graphs with Respect to the Cuts

• L. Sunil Chandran
• N. S. Narayanaswamy
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3106)

## Abstract

A cut (A,B) (where B = VA) in a graph G(V,E) is called internal, iff there exists a node x in A which is not adjacent to any node in B and there exists a node yB such that it is not adjacent to any node in A. In this paper, we present a theorem regarding the arrangement of cliques in a chordal graph with respect to its internal cuts. Our main result is that given any internal cut (A,B) in a chordal graph G, there exists a clique with κ(G) + 1 nodes (where κ(G) is the vertex connectivity of G) such that it is (approximately) bisected by the cut (A,B). In fact we give a stronger result: For any internal cut (A,B) of a chordal graph, for each i, 0 ≤ iκ(G) +1, there exists a clique Ki such that |Ki| = κ(G) + 1, |AKi| = i and |BKi| = κ(G) + 1 – i.

An immediate corollary of the above result is that the number of edges in any internal cut (of a chordal graph) should be Ω (k2), where κ(G) = k. Prompted by this observation, we investigate the size of internal cuts in terms of the vertex connectivity of the chordal graphs. As a corollary, we show that in chordal graphs, if the edge connectivity is strictly less than the minimum degree, then the size of the mincut is at least $$\frac{\kappa(G)(\kappa (G)+1)}{2}$$, where κ(G) denotes the vertex connectivity. In contrast, in a general graph the size of the mincut can be equal to κ(G). This result is tight.

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### References

1. 1.
Buneman, P.: A characterisation of rigid circuit graphs. Discrete Mathematics 9, 205–212 (1974)
2. 2.
Chandran, L.S.: Edge connectivity vs vertex connectivity in chordal graphs. In: Wang, J. (ed.) COCOON 2001. LNCS, vol. 2108, p. 384. Springer, Heidelberg (2001)
3. 3.
Sunil Chandran, L.: A linear time algorithm for enumerating all the minimum and minimal separators of a chordal graph. In: Wang, J. (ed.) COCOON 2001. LNCS, vol. 2108, pp. 308–317. Springer, Heidelberg (2001)
4. 4.
Chandran, L.S., Kavitha, T., Subramanian, C.R.: Isoperimetric inequalities and the width parameters of graphs. In: Warnow, T.J., Zhu, B. (eds.) COCOON 2003. LNCS, vol. 2697, pp. 385–395. Springer, Heidelberg (2003)
5. 5.
Chandrasekaran, R., Tamir, A.: Polynomially bounded algorithms for locating p-centres on a tree. Math. Programming 22, 304–315 (1982)
6. 6.
Chartrand, G., Harary, F.: Graphs with prescribed connectivities. In: Erdos, P., Katona, G. (eds.) Theory of Graphs, pp. 61–63. Akademiai Kiado, Budapest (1968)Google Scholar
7. 7.
Golumbic, M.C.: Algorithmic Graph Theory And Perfect Graphs. Academic Press, New York (1980)
8. 8.
Harary, F.: Graph Theory. Addison-Wesley, Reading (1969)Google Scholar
9. 9.
Kloks, T.: Treewidth. LNCS, vol. 842. Springer, Heidelberg (1994)
10. 10.
Mohring, R.H.: Graph Problems Related To Gate Matrix Layout And PLA Folding. In: Computational Graph Theory, pp. 17–52. Springer, Wein (1990)Google Scholar
11. 11.
Papadimitriou, C., Yannakakis, M.: Scheduling interval ordered tasks. SIAM Journal of Computing 8, 405–409 (1979)
12. 12.
Rose, D.: Triangulated graphs and the elimination process. J. Math. Ana. Appl. 32, 597–609 (1970)
13. 13.
Rose, D.: A graph theoretic study of the numerical solution of sparse positive definite systems of linear equations. In: Graph Theory and Computing, pp. 183–217. Academic Press, London (1972)Google Scholar
14. 14.
Whitney, H.: Congruent graphs and the connectivity of graphs. American J.Math 54, 150–168 (1932)
15. 15.
Yannakakis, M.: Computing the minimum Fill–in is NP–complete. SIAM J. on Alge. Discre. Math. 2, 77–79 (1981)

## Copyright information

© Springer-Verlag Berlin Heidelberg 2004

## Authors and Affiliations

• L. Sunil Chandran
• 1
• N. S. Narayanaswamy
• 2
1. 1.Max–Planck Institute for InformatikSaarbrückenGermany
2. 2.Department of Computer Science and EngineeringIndian Institute of TechnologyChennaiIndia