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## Discussiones Mathematicae Graph Theory

2002 | 22 | 1 | 159-172
Tytuł artykułu

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

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
159-172
Opis fizyczny
Daty
wydano
2002
otrzymano
2000-08-04
poprawiono
2001-12-23
Twórcy
autor
• Institut für Informatik, Universität zu Köln, D-50969 Köln, Germany
• Mathematics Department, Aalborg University, DK-9220 Aalborg Ø, Denmark
Bibliografia
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• [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, doi: 10.1016/S0012-365X(99)00251-4.
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• [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, doi: 10.1016/S0021-9800(70)80011-4.
• [15] M.D. Plummer, Well-covered graphs: a survey, Quaestiones Math. 16 (1993) 253-287, doi: 10.1080/16073606.1993.9631737.
• [16] B. Randerath and L. Volkmann, Characterization of graphs with equal domination and covering number, Discrete Math. 191 (1998) 159-169, doi: 10.1016/S0012-365X(98)00103-4.
• [17] R.S. Sankaranarayanan and L.K. Stewart, Complexity results for well-covered graphs, Networks 22 (1992) 247-262, doi: 10.1002/net.3230220304.
• [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.
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