Generating subgraphs in chordal graphs

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. 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 $WCW(G)$. 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 $WCW(G)$. Deciding whether an input graph $G$ is well-covered is co-NP-complete. Hence, 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. A graph is chordal if every induced cycle is a triangle. It is known that finding $WCW(G)$ 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.Comment: 13 pages, 1 figure. arXiv admin note: text overlap with arXiv:1401.029

Recognizing Generating Subgraphs in Graphs without Cycles of Lengths 6 and 7

Let $B$ be an induced complete bipartite subgraph of $G$ on vertex sets of bipartition $B_{X}$ and $B_{Y}$. The subgraph $B$ is {\it generating} if there exists an independent set $S$ such that each of $S \cup B_{X}$ and $S \cup B_{Y}$ is a maximal independent set in the graph. If $B$ is generating, it \textit{produces} the restriction $w(B_{X})=w(B_{Y})$. Let $w:V(G) \longrightarrow\mathbb{R}$ be a weight function. We say that $G$ is $w$-well-covered if all maximal independent sets are of the same weight. The graph $G$ is $w$-well-covered if and only if $w$ satisfies all restrictions produced by all generating subgraphs of $G$. Therefore, generating subgraphs play an important role in characterizing weighted well-covered graphs. It is an \textbf{NP}-complete problem to decide whether a subgraph is generating, even when the subgraph is isomorphic to $K_{1,1}$ \cite{bnz:related}. We present a polynomial algorithm for recognizing generating subgraphs for graphs without cycles of lengths 6 and 7.Comment: 13 pages, 3 figure

Recognizing generating subgraphs revisited

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. 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 as $WCW(G).$ Deciding whether an input graph $G$ is well-covered is co-NP-complete. Therefore, finding $WCW(G)$ is co-NP-hard. A generating subgraph of a graph $G$ is an induced complete bipartite subgraph $B$ of $G$ on vertex sets of bipartition $B_{X}$ and $B_{Y}$, such that each of $S \cup B_{X}$ and $S \cup B_{Y}$ is a maximal independent set of $G$, for some independent set $S$. If $B$ is generating, then $w(B_{X})=w(B_{Y})$ for every weight function $w \in WCW(G)$. Therefore, generating subgraphs play an important role in finding $WCW(G)$. The decision problem whether a subgraph of an input graph is generating is known to be NP-complete. In this article, we prove NP-completeness of the problem for graphs without cycles of length 3 and 5, and for bipartite graphs with girth at least 6. On the other and, we supply polynomial algorithms for recognizing generating subgraphs and finding $WCW(G)$, when the input graph is bipartite without cycles of length 6. We also present a polynomial algorithm which finds $WCW(G)$ when $G$ does not contain cycles of lengths 3, 4, 5, and 7.Comment: 22 pages, 2 figures. arXiv admin note: text overlap with arXiv:1401.029