Well-Covered Graphs With Constraints On $\Delta$ And $\delta$

Abstract

A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices, while the weight of a set of vertices is the sum of their weights. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space, denoted WCW(G). In what follows, all weights are real. Let B be a complete bipartite induced subgraph of G on vertex sets of bipartition $B_{X}$ and $B_{Y}$. Then B is generating if there exists an independent set S such that $S \cup B_{X}$ and $S \cup B_{Y}$ are both maximal independent sets of G. Generating subgraphs play an important role in finding WCW(G). In the restricted case that a generating subgraph B is isomorphic to $K_{1,1}$, its unique edge is called a relating edge. Deciding whether an input graph G is well-covered is co-NP-complete. Therefore finding WCW(G) is co-NP-hard. Deciding whether an edge is relating is NP-complete. Therefore, deciding whether a subgraph is generating is NP-complete as well. This article deals with graphs G such that $\Delta (G)=|V(G)|-k$ for some $k\in \mathbb {N}$. We prove that for this family recognizing well-covered graphs is a polynomial problem, while finding WCW(G) is co-NP-hard. To the best of our knowledge, this is the first family of graphs in the literature known to have these properties. For this set of graphs, recognizing relating edges and generating subgraphs is NP-complete. The article also deals with connected graphs for which $\delta (G) = k$ or $\delta (G) \ge \frac{k-1}{k}|V(G)|$. For these families of graphs recognizing well-covered graphs is co-NP-complete, while recognizing relating edges is NP-complete.