Generating subgraphs in chordal graphs

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 the vertex set of $G$. 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 $WCW\left(G\right)$.

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 $G$. In the restricted case that a generating subgraph $B$ is isomorphic to $K_{1,1}$, the unique edge in $B$ is called a relating edge. Generating subgraphs play an important role in finding $WCW\left(G\right)$.

Deciding whether an input graph $G$ is well-covered is co-NP-complete. Hence, finding $WCW\left(G\right)$ 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.

A graph is chordal if every induced cycle is a triangle. It is known that finding $WCW\left(G\right)$ can be done polynomially in the restricted case that $G$ is chordal. Thus, recognizing well-covered chordal graphs is a polynomial problem. We present a polynomial algorithm for recognizing relating edges and generating subgraphs in chordal graphs.