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Distance-hereditary graphs are clique-perfect. (English) Zbl 1110.68108
Summary: We show that the clique-transversal number $$\tau_C(G)$$ and the clique-independence number $$\alpha_C(G)$$ are equal for any distance-hereditary graph $$G$$. As a byproduct of proving that $$\tau_C(G)= \alpha_C(G)$$, we give a linear-time algorithm to find a minimum clique-transversal set and a maximum clique-independent set simultaneously for distance-hereditary graphs.
Reviewer: Reviewer (Berlin)

##### MSC:
 68R10 Graph theory (including graph drawing) in computer science 05C12 Distance in graphs 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05C85 Graph algorithms (graph-theoretic aspects)
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##### References:
 [1] Andreae, T., On the clique-transversal number of chordal graphs, Discrete math., 191, 3-11, (1998) · Zbl 0955.05058 [2] Andreae, T.; Flotow, C., On covering all cliques of a chordal graph, Discrete math., 149, 299-302, (1996) · Zbl 0846.05050 [3] Andreae, T.; Schughart, M.; Tuza, Zs., Clique transversal sets of line graphs and complements of line graphs, Discrete math., 88, 11-20, (1991) · Zbl 0734.05077 [4] Balachandran, V.; Nagavamsi, P.; Rangan, C.P., Clique transversal and clique independence on comparability graphs, Inform. process. lett., 58, 181-184, (1996) [5] Bandelt, H.J.; Mulder, H.M., Distance hereditary graphs, J. combin. theory ser. B, 41, 182-208, (1986) · Zbl 0605.05024 [6] Brandstädt, A.; Chepoi, V.D.; Dragan, F.F., Clique $$r$$-domination and clique $$r$$-packing problems on dually chordal graphs, SIAM J. discrete math., 10, 1, 109-127, (1997) · Zbl 0869.05048 [7] Brandstädt, A.; Le, V.B.; Spinrad, J.P., Graph classes—A survey, SIAM monographs on discrete mathematics and applications, (1999), SIAM Philadelphia, PA [8] Chang, G.J.; Farber, M.; Tuza, Zs., Algorithmic aspects of neighborhood numbers, SIAM J. discrete math., 6, 24-29, (1993) · Zbl 0777.05085 [9] Chang, M.S.; Chen, Y.H.; Chang, G.J.; Yan, J.H., Algorithmic aspects of the generalised clique transversal problem on chordal graphs, Discrete appl. math., 66, 189-203, (1996) · Zbl 0854.68072 [10] M.S. Chang, S.Y. Hsieh, G.H. Chen, Dynamic Programming on Distance-Hereditary Graphs, Lecture Notes in Computer Science, vol. 1350, 1997, Springer, Berlin, pp. 344-353. [11] M.S. Chang, T. Kloks, C.M. Lee, Maximum clique transversals, in: Proceedings of the 27th International Workshop on Graph-Theoretic Concepts in Computer Science, Lecture Notes in Computer Science, vol. 2204, 2001, Springer, Berlin, pp. 32-43. · Zbl 1042.68619 [12] Courcelle, B.; Engelfriet, J.; Rozenberg, G., Handle-rewriting hypergraph grammars, J. comput. system sci., 46, 218-270, (1993) · Zbl 0825.68446 [13] Courcelle, B.; Olariu, S., Upper bounds to the clique-width of graphs, Discrete appl. math., 101, 77-114, (2000) · Zbl 0958.05105 [14] Dahlhaus, E.; Manuel, P.D.; Miller, M., Maximum $$h$$-colourable subgraph problem in balanced graphs, Inform. process. lett., 65, 301-303, (1998) · Zbl 1338.68098 [15] Durán, G.; Lin, M.C.; Szwarcfiter, J.L., On clique-transversal and clique-independent sets, Ann. oper. res., 116, 71-77, (2002) · Zbl 1013.90107 [16] Eades, P.; Keil, M.; Manuel, P.D.; Miller, M., Two minimum dominating sets with minimum intersection in chordal graphs, Nordic J. comput., 3, 220-237, (1996) [17] Erdös, P.; Gallai, T.; Tuza, Zs., Covering the cliques of a graph with vertices, Discrete math., 108, 279-289, (1992) · Zbl 0766.05063 [18] Golumbicm, M.C.; Rotics, U., On the clique-width of some perfect graph classes, Int. J. foundations comput. sci., 11, 3, 423-443, (2000) · Zbl 1320.05090 [19] Guruswami, V.; Rangan, C.P., Algorithmic aspects of clique-transversal and clique-independent sets, Discrete appl. math., 100, 183-202, (2000) · Zbl 0948.68135 [20] Hammer, P.L.; Maffray, F., Completely separable graphs, Discrete appl. math., 27, 85-90, (1990) · Zbl 0694.05060 [21] C.M. Lee, M.S. Chang, S.C. Sheu, The clique transversal and clique independence of distance hereditary graphs, in: Proceedings of the 19th Workshop on Combinatorial Mathematics and Computation Theory, Taiwan, 2002, pp. 64-69. [22] Lehel, J.; Tuza, Zs., Neighborhood perfect graphs, Discrete math., 61, 93-101, (1986) · Zbl 0602.05053 [23] Lonc, Z.; Rival, I., Chains, antichains and fibres, J. combin. theory ser. A, 44, 207-228, (1987) · Zbl 0637.06001 [24] S.C. Sheu, The weighted clique transversal set problem on distance-hereditary graphs, Master Thesis, Department of Computer Science and Information Engineering, Chung Cheng University, Taiwan, 2001. [25] Tuza, Zs., Covering all cliques of a graph, Discrete math., 86, 117-126, (1990) · Zbl 0744.05040
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