On well-covered graphs of odd girth 7 or greater

Bert Randerath ; Preben Dahl Vestergaard

Discussiones Mathematicae Graph Theory, Tome 22 (2002), p. 159-172 / Harvested from The Polish Digital Mathematics Library

Résumé

A maximum independent set of vertices in a graph is a set of pairwise nonadjacent vertices of largest cardinality α. Plummer [14] defined a graph to be well-covered, if every independent set is contained in a maximum independent set of G. One of the most challenging problems in this area, posed in the survey of Plummer [15], is to find a good characterization of well-covered graphs of girth 4. We examine several subclasses of well-covered graphs of girth ≥ 4 with respect to the odd girth of the graph. We prove that every isolate-vertex-free well-covered graph G containing neither C₃, C₅ nor C₇ as a subgraph is even very well-covered. Here, a isolate-vertex-free well-covered graph G is called very well-covered, if G satisfies α(G) = n/2. A vertex set D of G is dominating if every vertex not in D is adjacent to some vertex in D. The domination number γ(G) is the minimum order of a dominating set of G. Obviously, the inequality γ(G) ≤ α(G) holds. The family $_{γ=α}$ of graphs G with γ(G) = α(G) forms a subclass of well-covered graphs. We prove that every connected member G of $_{γ=α}$ containing neither C₃ nor C₅ as a subgraph is a K₁, C₄,C₇ or a corona graph.

Détail

Publié le : 2002-01-01

Zbl 1012.05123

EUDML-ID : urn:eudml:doc:270191

Bibliographie

[000] [1] M.O. Albertson, L. Chan and R. Haas, Independence and graph homomorphisms, J. Graph Theory 17 (1993) 581-588, doi: 10.1002/jgt.3190170503. | Zbl 0781.05028

[001] [2] X. Baogen, E. Cockayne, T.W. Haynes, S.T. Hedetniemi and Z. Shangchao, Extremal graphs for inequalities involving domination parameters, Discrete Math. 216 (2000) 1-10, doi: 10.1016/S0012-365X(99)00251-4. | Zbl 0954.05037

[002] [3] C. Berge, Regularizable graphs, Ann. Discrete Math. 3 (1978) 11-19, doi: 10.1016/S0167-5060(08)70493-X.

[003] [4] V. Chvátal and P.J. Slater, A note on well-covered graphs, Ann. Discrete Math. 55 (1993) 179-182, doi: 10.1016/S0167-5060(08)70387-X. | Zbl 0801.68119

[004] [5] O. Favaron, Very well-covered graphs, Discrete Math. 42 (1982) 177-187, doi: 10.1016/0012-365X(82)90215-1. | Zbl 0507.05053

[005] [6] A. Finbow and B. Hartnell, A game related to covering by stars, Ars Combin. 16 (A) (1983) 189-198.

[006] [7] A. Finbow, B. Hartnell and R.J. Nowakowski, A characterization of well-covered graphs of girth 5 or greater, J. Combin. Theory (B) 57 (1993) 44-68, doi: 10.1006/jctb.1993.1005. | Zbl 0777.05088

[007] [8] A. Finbow, B. Hartnell and R.J. Nowakowski, A characterization of well-covered graphs which contain neither 4- nor 5-cycles, J. Graph Theory 18 (1994) 713-721, doi: 10.1002/jgt.3190180707. | Zbl 0810.05058

[008] [9] J.F. Fink, M.S. Jacobson, L.F. Kinch and J. Roberts, On graphs having domination number half their order, Period. Math. Hungar. 16 (1985) 287-293, doi: 10.1007/BF01848079. | Zbl 0602.05043

[009] [10] O. Ore, Theory of Graphs (Amer. Math. Soc. Colloq. Publ. 38, 1962).

[010] [11] C. Payan and N.H. Xuong, Domination-balanced graphs. J. Graph Theory 6 (1982) 23-32, doi: 10.1002/jgt.3190060104. | Zbl 0489.05049

[011] [12] M.R. Pinter, A class of planar well-covered graphs with girth four, J. Graph Theory 19 (1995) 69-81, doi: 10.1002/jgt.3190190108. | Zbl 0813.05054

[012] [13] M.R. Pinter, A class of well-covered graphs with girth four, Ars Combin. 45 (1997) 241-255. | Zbl 0933.05119

[013] [14] M.D. Plummer, Some covering concepts in graphs, J. Combin. Theory 8 (1970) 91-98, doi: 10.1016/S0021-9800(70)80011-4. | Zbl 0195.25801

[014] [15] M.D. Plummer, Well-covered graphs: a survey, Quaestiones Math. 16 (1993) 253-287, doi: 10.1080/16073606.1993.9631737. | Zbl 0817.05068

[015] [16] B. Randerath and L. Volkmann, Characterization of graphs with equal domination and covering number, Discrete Math. 191 (1998) 159-169, doi: 10.1016/S0012-365X(98)00103-4. | Zbl 0955.05088

[016] [17] R.S. Sankaranarayanan and L.K. Stewart, Complexity results for well-covered graphs, Networks 22 (1992) 247-262, doi: 10.1002/net.3230220304. | Zbl 0780.90104

[017] [18] J. Staples, Ph. D. dissertation (Vanderbilt University, Nashville, TN, 1975).

[018] [19] L. Szamkołowicz, Sur la classification des graphes en vue des propriétés de leurs noyaux, Prace Nauk. Inst. Mat. i Fiz. Teoret., Politechn. Wrocław., Ser. Stud. Mater. 3 (1970) 15-21. | Zbl 0264.05125

[019] [20] J. Topp and L. Volkmann, On domination and independence numbers of graphs, Resultate Math. 17 (1990) 333-341. | Zbl 0748.05066

[020] [21] W.T. Tutte, The 1-factors of oriented graphs, Proc. Amer. Math. Soc. 4 (1953) 922-931. | Zbl 0052.20004