Geodesic
On well-covered graphs of odd girth 7 or greater
Discussiones Mathematicae. Graph Theory, Tome 22 (2002) no. 1, pp. 159-172.

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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 _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.
Keywords: well-covered, independence number, domination number, odd girth 
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Randerath, Bert; Vestergaard, Preben. On well-covered graphs of odd girth 7 or greater. Discussiones Mathematicae. Graph Theory, Tome 22 (2002) no. 1, pp. 159-172. http://geodesic.mathdoc.fr/item/DMGT_2002_22_1_a12/

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