A maximum priority matching is a matching in an undirected graph that maximizes a priority score defined with respect to given vertex priorities. An earlier paper showed how to find maximum priority matchings in unweighted graphs. This paper describes an algorithm for bipartite graphs that is faster when the number of distinct priority classes is limited. For graphs with k distinct priority cla...

An induced matching M of a graph G is a set of pairwise nonadjacent edges such that no two edges of M are joined by an edge in G. The problem of /nding a maximum induced matching is known to be NP-hard even for bipartite graphs of maximum degree four. In this paper, we study the maximum induced matching problem on classes of graphs related to AT-free graphs. We /rst de/ne a wider class of graph...

For an arbitrary tree we investigate the problems of constructing a maximum matching which minimizes or maximizes the cardinality of a maximum matching of the graph obtained from original one by its removal and present corresponding polynomial algorithms.

For bipartite graphs the NP-completeness is proved for the problem of existence of maximum matching which removal leads to a graph with given lower(upper) bound for the cardinality of its maximum matching.

The maximum matching width is a graph width parameter that is defined on a branchdecomposition over the vertex set of a graph. In this short paper, we prove that the problem of computing the maximum matching width is NP-hard.