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Discussiones Mathematicae Graph Theory 22(1) (2002) 159-172
DOI: https://doi.org/10.7151/dmgt.1165

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 G_{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 G_{γ = 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.

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Received 4 August 2000
Revised 23 December 2001