Discussiones Mathematicae Graph Theory 22(1) (2002) 159-172

ON WELL-COVERED GRAPHS OF ODD GIRTH 7 OR GREATER

Bert Randerath

Institut für Informatik
Universität zu Köln
D-50969 Köln, Germany
e-mail: randerath@informatik.uni-koeln.de

 Preben Dahl Vestergaard

Mathematics Department
Aalborg University
DK-9220 Aalborg Ø, Denmark
e-mail: pdv@math.auc.dk

Abstract

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 Gγ = α of graphs G with γ(G) = α(G) forms a subclass of well-covered graphs. We prove that every connected member G of Gγ = α containing neither C3 nor C5 as a subgraph is a K1, C4,C7 or a corona graph.

Keywords: well-covered, independence number, domination number, odd girth.

2000 Mathematics Subject Classification: 05C70, 05C75.

References

[1] M.O. Albertson, L. Chan and R. Haas, Independence and graph homomorphisms, J. Graph Theory 17 (1993) 581-588.
[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.
[3] C. Berge, Regularizable graphs, Ann. Discrete Math. 3 (1978) 11-19.
[4] V. Chvátal and P.J. Slater, A note on well-covered graphs, Ann. Discrete Math. 55 (1993) 179-182.
[5] O. Favaron, Very well-covered graphs, Discrete Math. 42 (1982) 177-187.
[6] A. Finbow and B. Hartnell, A game related to covering by stars, Ars Combin. 16 (A) (1983) 189-198.
[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.
[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.
[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.
[10] O. Ore, Theory of Graphs (Amer. Math. Soc. Colloq. Publ. 38, 1962).
[11] C. Payan and N.H. Xuong, Domination-balanced graphs. J. Graph Theory 6 (1982) 23-32.
[12] M.R. Pinter, A class of planar well-covered graphs with girth four, J. Graph Theory 19 (1995) 69-81.
[13] M.R. Pinter, A class of well-covered graphs with girth four, Ars Combin. 45 (1997) 241-255.
[14] M.D. Plummer, Some covering concepts in graphs, J. Combin. Theory 8 (1970) 91-98.
[15] M.D. Plummer, Well-covered graphs: a survey, Quaestiones Math. 16 (1993) 253-287.
[16] B. Randerath and L. Volkmann, Characterization of graphs with equal domination and covering number, Discrete Math. 191 (1998) 159-169.
[17] R.S. Sankaranarayanan and L.K. Stewart, Complexity results for well-covered graphs, Networks 22 (1992) 247-262.
[18] J. Staples, Ph. D. dissertation (Vanderbilt University, Nashville, TN, 1975).
[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.
[20] J. Topp and L. Volkmann, On domination and independence numbers of graphs, Resultate Math. 17 (1990) 333-341.
[21] W.T. Tutte, The 1-factors of oriented graphs, Proc. Amer. Math. Soc. 4 (1953) 922-931.

Received 4 August 2000
Revised 23 December 2001