A Characterization of the 1-well-covered Graphs with no 4-cycles

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.