Abstract:
A subset M ⊆ E of edges of a graph G = (V, E) is called a matching in G if no
two edges in M share a common vertex. A matching M in G is called an induced
matching if G[M], the subgraph of G induced by M, is the same as G[S], the subgraph
of G induced by S = {v ∈ V| v is incident on an edge of M}. The Maximum
Induced Matching problem is to find an induced matching of maximum cardinality.
Given a graph G and a positive integer k, the Induced Matching Decision
problem is to decide whether G has an induced matching of cardinality at least k. The
Maximum Weight Induced Matching problem in a weighted graph G = (V, E)
in which the weight of each edge is a positive real number, is to find an induced
matching such that the sum of the weights of its edges is maximum. It is known
that the Induced Matching Decision problem and hence the Maximum Weight
Induced Matching problem is known to be NP-complete for general graphs and
bipartite graphs. In this paper,we strengthened this result by showing that the Induced
Matching Decision problem is NP-complete for star-convex bipartite graphs, combconvex
bipartite graphs, and perfect elimination bipartite graphs, the subclasses of the
class of bipartite graphs. On the positive side, we propose polynomial time algorithms
for the Maximum Weight Induced Matching problem for circular-convex bipartite
graphs and triad-convex bipartite graphs by making polynomial time reductions
from the Maximum Weight Induced Matching problem in these graph classes
to the Maximum Weight Induced Matching problem in convex bipartite graphs.
