5 research outputs found
On the Parameterized Complexity of the Acyclic Matching Problem
A matching is a set of edges in a graph with no common endpoint. A matching M
is called acyclic if the induced subgraph on the endpoints of the edges in M is
acyclic. Given a graph G and an integer k, Acyclic Matching Problem seeks for
an acyclic matching of size k in G. The problem is known to be NP-complete. In
this paper, we investigate the complexity of the problem in different aspects.
First, we prove that the problem remains NP-complete for the class of planar
bipartite graphs of maximum degree three and arbitrarily large girth. Also, the
problem remains NP-complete for the class of planar line graphs with maximum
degree four. Moreover, we study the parameterized complexity of the problem. In
particular, we prove that the problem is W[1]-hard on bipartite graphs with
respect to the parameter k. On the other hand, the problem is fixed parameter
tractable with respect to the parameters tw and (k, c4), where tw and c4 are
the treewidth and the number of cycles with length 4 of the input graph. We
also prove that the problem is fixed parameter tractable with respect to the
parameter k for the line graphs and every proper minor-closed class of graphs
(including planar graphs)
-matchings Parameterized by Treewidth
A \emph{matching} is a subset of edges in a graph that do not share an
endpoint. A matching is a \emph{-matching} if the subgraph of
induced by the endpoints of the edges of satisfies property
. For example, if the property is that of being a
matching, being acyclic, or being disconnected, then we obtain an \emph{induced
matching}, an \emph{acyclic matching}, and a \emph{disconnected matching},
respectively. In this paper, we analyze the problems of the computation of
these matchings from the viewpoint of Parameterized Complexity with respect to
the parameter \emph{treewidth}.Comment: To Appear in the proceedings of WG 202