# On well-covered graphs of odd girth 7 or greater

• Volume: 22, Issue: 1, page 159-172
• ISSN: 2083-5892

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## Abstract

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

## How to cite

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Bert Randerath, and Preben Dahl Vestergaard. "On well-covered graphs of odd girth 7 or greater." Discussiones Mathematicae Graph Theory 22.1 (2002): 159-172. <http://eudml.org/doc/270191>.

@article{BertRanderath2002,
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 $_\{γ=α\}$ 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.},
author = {Bert Randerath, Preben Dahl Vestergaard},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {well-covered; independence number; domination number; odd girth; well-covered graphs},
language = {eng},
number = {1},
pages = {159-172},
title = {On well-covered graphs of odd girth 7 or greater},
url = {http://eudml.org/doc/270191},
volume = {22},
year = {2002},
}

TY - JOUR
AU - Bert Randerath
AU - Preben Dahl Vestergaard
TI - On well-covered graphs of odd girth 7 or greater
JO - Discussiones Mathematicae Graph Theory
PY - 2002
VL - 22
IS - 1
SP - 159
EP - 172
AB - 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.
LA - eng
KW - well-covered; independence number; domination number; odd girth; well-covered graphs
UR - http://eudml.org/doc/270191
ER -

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