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,...

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Bibliographic Details
Main Author Tankus, David
Format Journal Article
LanguageEnglish
Published 30.08.2018
Subjects
Computer Science - Computational Complexity
Computer Science - Discrete Mathematics
Mathematics - Combinatorics
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Summary: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.
DOI:10.48550/arxiv.1808.10137