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 G of order n ( G ) without an isolated vertex, Claude Berge (C. Berge, Some common properties for regularizable graphs, edge-critical graphs and b -graphs, Ann. Discrete Math. 12 (1982) 31–44) proved that the independence number α ( G ) of G 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). end these two results in two different ways. First, we study the structure and recognition of the well-covered graphs G without an isolated vertex that have independence number n ( G ) − k 2 for some non-negative integer k . For k = 1 , we give a complete structural description of these graphs, and for a general but fixed k , we describe a polynomial time recognition algorithm. Second, we relax the condition of well-coveredness and consider graphs G without an isolated vertex for which the independence number α ( G ) and the independent domination number i ( G ) satisfy α ( G ) − i ( G ) ≤ k for some non-negative integer k . 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.