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2002 | 22 | 1 | 159-172
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On well-covered graphs of odd girth 7 or greater

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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.
Opis fizyczny
  • Institut für Informatik, Universität zu Köln, D-50969 Köln, Germany
  • Mathematics Department, Aalborg University, DK-9220 Aalborg Ø, Denmark
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