Extending Berge’s and Favaron’s results about well-covered graphs

A graph is well-covered if all its maximal independent sets have the same order. For a well-covered graph GG of order n(G)n(G) without an isolated vertex, Claude Berge (C. Berge, Some common properties for regularizable graphs, edge-critical graphs and bb-graphs, Ann. Discrete Math. 12 (1982) 31–44) proved that the independence number α(G)α(G) of GG is at most n(G)2. The extremal graphs for this result are known as the very well-covered graphs and were characterized by Odile Favaron (O. Favaron, Very well-covered graphs, Discrete Math. 42 (1982) 177–187).We extend these two results in two different ways. First, we study the structure and recognition of the well-covered graphs GG without an isolated vertex that have independence number n(G)−k2 for some non-negative integer kk. For k=1k=1, we give a complete structural description of these graphs, and for a general but fixed kk, we describe a polynomial time recognition algorithm. Second, we relax the condition of well-coveredness and consider graphs GG without an isolated vertex for which the independence number α(G)α(G) and the independent domination number i(G)i(G) satisfy α(G)−i(G)≤kα(G)−i(G)≤k for some non-negative integer kk. We prove a suitable version of Berge’s result for these graphs, derive an upper bound on the independence number as a corollary, and discuss its relation to Favaron’s characterization.