The class of cographs, or complement-reducible graphs, arises naturally in many different areas of applied mathematics and computer science. We show that the problem of finding a maximum matching in a cograph can be solved optimally in parallel by reducing it to parenthesis matching. With an $n$-vertex cograph $G$ represented by its parse tree as input, our algorithm finds a maximum matching in...

We consider complete graphs with nonnegative edge weights. A p-matching is a set of p disjoint edges. We prove the existence of a maximal (with respect to inclusion) matching M that contains for any p ≤ |M | p edges whose total weight is at least 1 √ 2 of the maximum weight of a p-matching. We use this property to approximate the metric maximum clustering problem with given cluster sizes.

In this paper we deal with simple finite undirected graphs (no loops and no multiple edges). G = (V, E) denotes a graph with vertex set V and edge set E. A matching of a graph is a set of nonadjacent edges, and a maximum matching, denoted by M(G), of G is one of maximum cardinality. n(G) denotes the number of unsaturated vertices (i.e., vertices with which no edge of a matching is incident) in ...

For a graph G let L(G) and l(G) denote the size of the largest and smallest maximum matching of graphs obtained from G by removing its maximum matchings. We show that L(G) ≤ 2l(G), and L(G) ≤ 3 2 l(G) provided that G contains a perfect matching. We also characterize the class of graphs for with L(G) = 2l(G). Our characterization implies the existence of a polynomial algorithm for testing the pr...

Let D=(V,A) be a digraphs without isolated vertices. The first Zagreb index of digraph D is defined as summation over all arcs, M1(D)=12∑uv∈A(du++dv−), where du+(resp. du−) denotes the out-degree (resp. in-degree) vertex u. In this paper, we give maximal values and set orientations unicyclic graphs with n vertices matching number m (2≤m≤⌊n2⌋).