Abstract
The problem of determining which graphs have the property that every maximal independent set of vertices is also a maximum independent set was proposed in 1970 by M.D. Plummer who called such graphs well-covered. Whereas determining the independence number of an arbitrary graph is NP-complete, for a well-covered graph one can simply apply the greedy algorithm. A well-covered graph G is 1-well-covered if and only if, for every vertex v in G, G — v is also well covered and has the same independence number. The notion of a 1-well-covered graph was introduced by J. Staples in her 1975 dissertation and was further investigated by M. Pinter in 1991 and later. In this note the 1-well-covered graphs with no 4-cycles are characterized.
Research supported in part by NSERC of Canada.
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© 2006 Birkhäuser Verlag Basel/Switzerland
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Hartnell, B.L. (2006). A Characterization of the 1-well-covered Graphs with no 4-cycles. In: Bondy, A., Fonlupt, J., Fouquet, JL., Fournier, JC., Ramírez Alfonsín, J.L. (eds) Graph Theory in Paris. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7400-6_17
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