3 research outputs found
Generating subgraphs in chordal graphs
A graph is well-covered if all its maximal independent sets are of the
same cardinality. Assume that a weight function is defined on its vertices.
Then is -well-covered if all maximal independent sets are of the same
weight. For every graph , the set of weight functions such that is
-well-covered is a vector space, denoted . Let be a complete
bipartite induced subgraph of on vertex sets of bipartition and
. Then is generating if there exists an independent set such
that and are both maximal independent sets of
. In the restricted case that a generating subgraph is isomorphic to
, the unique edge in is called a relating edge. Generating
subgraphs play an important role in finding . Deciding whether an input
graph is well-covered is co-NP-complete. Hence, finding 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
can be done polynomially in the restricted case that 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 be an induced complete bipartite subgraph of on vertex sets of
bipartition and . The subgraph is {\it generating} if there
exists an independent set such that each of and is a maximal independent set in the graph. If is generating, it
\textit{produces} the restriction . Let be a weight function. We say that is
-well-covered if all maximal independent sets are of the same weight. The
graph is -well-covered if and only if satisfies all restrictions
produced by all generating subgraphs of . 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 \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 is well-covered if all its maximal independent sets are of the
same cardinality. Assume that a weight function is defined on its vertices.
Then is -well-covered if all maximal independent sets are of the same
weight. For every graph , the set of weight functions such that is
-well-covered is a vector space, denoted as Deciding whether an
input graph is well-covered is co-NP-complete. Therefore, finding
is co-NP-hard. A generating subgraph of a graph is an induced complete
bipartite subgraph of on vertex sets of bipartition and
, such that each of and is a maximal
independent set of , for some independent set . If is generating,
then for every weight function . Therefore,
generating subgraphs play an important role in finding . 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 , when the input graph is bipartite
without cycles of length 6. We also present a polynomial algorithm which finds
when 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