2 research outputs found
On Efficient Domination for Some Classes of -Free Bipartite Graphs
A vertex set in a finite undirected graph is an {\em efficient
dominating set} (\emph{e.d.s.}\ for short) of if every vertex of is
dominated by exactly one vertex of . The \emph{Efficient Domination} (ED)
problem, which asks for the existence of an e.d.s.\ in , is known to be
\NP-complete even for very restricted -free graph classes such as for
-free chordal graphs while it is solvable in polynomial time for
-free graphs. Here we focus on -free bipartite graphs: We show that
(weighted) ED can be solved in polynomial time for -free bipartite graphs
when is or for fixed , and similarly for -free
bipartite graphs with vertex degree at most 3, and when is .
Moreover, we show that ED is \NP-complete for bipartite graphs with diameter at
most 6.Comment: arXiv admin note: text overlap with arXiv:1701.0341
Finding Efficient Domination for -Free Bipartite Graphs in Polynomial Time
A vertex set in a finite undirected graph is an {\em efficient
dominating set} (\emph{e.d.s.}\ for short) of if every vertex of is
dominated by exactly one vertex of . The \emph{Efficient Domination} (ED)
problem, which asks for the existence of an e.d.s.\ in , is \NP-complete for
various -free bipartite graphs, e.g., Lu and Tang showed that ED is
\NP-complete for chordal bipartite graphs and for planar bipartite graphs;
actually, ED is \NP-complete even for planar bipartite graphs with vertex
degree at most 3 and girth at least for every fixed . Thus, ED is
\NP-complete for -free bipartite graphs and for -free bipartite
graphs.
In this paper, we show that ED can be solved in polynomial time for
-free bipartite graphs