COCOA 2010: Combinatorial Optimization and Applications pp 207-218 | Cite as
New Min-Max Theorems for Weakly Chordal and Dually Chordal Graphs
Abstract
A distance-k matching in a graph G is matching M in which the distance between any two edges of M is at least k. A distance-2 matching is more commonly referred to as an induced matching. In this paper, we show that when G is weakly chordal, the size of the largest induced matching in G is equal to the minimum number of co-chordal subgraphs of G needed to cover the edges of G, and that the co-chordal subgraphs of a minimum cover can be found in polynomial time. Using similar techniques, we show that the distance-k matching problem for k > 1 is tractable for weakly chordal graphs when k is even, and is NP-hard when k is odd. For dually chordal graphs, we use properties of hypergraphs to show that the distance-k matching problem is solvable in polynomial time whenever k is odd, and NP-hard when k is even. Motivated by our use of hypergraphs, we define a class of hypergraphs which lies strictly in between the well studied classes of acyclic hypergraphs and normal hypergraphs.
Preview
Unable to display preview. Download preview PDF.
References
- 1.Abuieda, A., Busch, A., Sritharan, R.: A min-max property of chordal bipartite graphs with applications. Graphs Combin. 26(3), 301–313 (2010)MathSciNetCrossRefMATHGoogle Scholar
- 2.Berge, C.: Hypergraphs. North-Holland Mathematical Library, vol. 45. North-Holland Publishing Co., Amsterdam (1989); Combinatorics of finite sets, Translated from the FrenchMATHGoogle Scholar
- 3.Brandstädt, A., Chepoi, V.D., Dragan, F.F.: The algorithmic use of hypertree structure and maximum neighbourhood orderings. Discrete Appl. Math. 82(1-3), 43–77 (1998)MathSciNetCrossRefMATHGoogle Scholar
- 4.Brandstädt, A., Dragan, F., Chepoi, V., Voloshin, V.: Dually chordal graphs. SIAM J. Discrete Math. 11(3), 437–455 (1998) (electronic)MathSciNetCrossRefMATHGoogle Scholar
- 5.Brandstädt, A., Dragan, F.F., Xiang, Y., Yan, C.: Generalized powers of graphs and their algorithmic use. In: Arge, L., Freivalds, R. (eds.) SWAT 2006. LNCS, vol. 4059, pp. 423–434. Springer, Heidelberg (2006)Google Scholar
- 6.Brandstädt, A., Van Bang, L., Spinrad, J.P.: Graph classes: a survey. SIAM Monographs on Discrete Mathematics and Applications. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1999)Google Scholar
- 7.Brandstädt, A., Mosca, R.: On distance-3 matchings and induced matchings. In: Lipshteyn, M., Levit, V.E., McConnell, R.M. (eds.) Graph Theory. LNCS, vol. 5420, pp. 116–126. Springer, Heidelberg (2009)CrossRefGoogle Scholar
- 8.Cameron, K.: Induced matchings. Discrete Appl. Math. 24(1-3), 97–102 (1989); First Montreal Conference on Combinatorics and Computer Science (1987)MathSciNetCrossRefMATHGoogle Scholar
- 9.Cameron, K., Sritharan, R., Tang, Y.: Finding a maximum induced matching in weakly chordal graphs. Discrete Math. 266(1-3), 133–142 (2003); The 18th British Combinatorial Conference, Brighton (2001)MathSciNetCrossRefMATHGoogle Scholar
- 10.Dirac, G.A.: On rigid circuit graphs. Abh. Math. Sem. Univ. Hamburg 25, 71–76 (1961)MathSciNetCrossRefMATHGoogle Scholar
- 11.Dragan, F.F., Prisakar’, K.F., Chepoĭ, V.D.: The location problem on graphs and the Helly problem. Diskret. Mat. 4(4), 67–73 (1992)MathSciNetMATHGoogle Scholar
- 12.Eschen, E., Sritharan, R.: A characterization of some graph classes with no long holes. J. Combin. Theory Ser. B 65(1), 156–162 (1995)MathSciNetCrossRefMATHGoogle Scholar
- 13.Hayward, R., Hoàng, C., Maffray, F.: Optimizing weakly triangulated graphs. Graphs Combin. 5(4), 339–349 (1989)MathSciNetCrossRefMATHGoogle Scholar
- 14.Hayward, R.B., Spinrad, J., Sritharan, R.: Weakly chordal graph algorithms via handles. In: Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, CA, pp. 42–49. ACM, New York (2000)Google Scholar
- 15.Kratochvíl, J., Tuza, Z.: Intersection dimensions of graph classes. Graphs Combin. 10(2), 159–168 (1994)MathSciNetCrossRefMATHGoogle Scholar
- 16.Ma, T.H., Spinrad, J.P.: On the 2-chain subgraph cover and related problems. J. Algorithms 17(2), 251–268 (1994)MathSciNetCrossRefMATHGoogle Scholar
- 17.McKee, T.A., Scheinerman, E.R.: On the chordality of a graph. J. Graph Theory 17(2), 221–232 (1993)MathSciNetCrossRefMATHGoogle Scholar
- 18.Moscarini, M.: Doubly chordal graphs, Steiner trees, and connected domination. Networks 23(1), 59–69 (1993)MathSciNetCrossRefMATHGoogle Scholar
- 19.Spinrad, J., Sritharan, R.: Algorithms for weakly triangulated graphs. Discrete Appl. Math. 59(2), 181–191 (1995)MathSciNetCrossRefMATHGoogle Scholar
- 20.Szwarcfiter, J.L., Bornstein, C.F.: Clique graphs of chordal and path graphs. SIAM J. Discrete Math. 7(2), 331–336 (1994)MathSciNetCrossRefMATHGoogle Scholar
- 21.Tarjan, R.E., Yannakakis, M.: Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM J. Comput. 13(3), 566–579 (1984)MathSciNetCrossRefMATHGoogle Scholar
- 22.Woodroofe, R.: Personal CommunicationGoogle Scholar
- 23.Yannakakis, M.: The complexity of the partial order dimension problem. SIAM J. Algebraic Discrete Methods 3(3), 351–358 (1982)MathSciNetCrossRefMATHGoogle Scholar
- 24.Yu, C.-W., Chen, G.-H., Ma, T.-H.: On the complexity of the k-chain subgraph cover problem. Theoret. Comput. Sci. 205(1-2), 85–98 (1998)MathSciNetCrossRefMATHGoogle Scholar