A matching in a graph G is a collection of non-intersecting edges. The matching is induced if no two edges in the matching are joined by an edge in G. This paper studies the complexity of nding a largest induced matching when the input graph is a tree. We describe the rst linear time algorithm for this problem.

Let G be a finite simple graph on the vertex set $$V(G) = \{x_{1}, \ldots , x_{n}\}$$ and match(G), min-match(G) ind-match(G) matching number, minimum number induced of G, respectively. $$K[V(G)] K[x_{1}, x_{n}]$$ denote polynomial ring over field K $$I(G) \subset K[V(G)]$$ edge ideal G. The relationship between these graph-theoretic invariants ring-theoretic quotient K[V(G)]/I(G) has been stud...

Yu, X., Cyclomatic numbers of connected induced subgraphs, Discrete Mathematics 105 (1992) 275-284. We give an upper bound for w(A), the minimum cyclomatic number of connected induced subgraphs containing a given independent set A of vertices in a given graph G. We also give an upper bound for w(A) when G is triangle-free. We show that these two bounds are best possible. Similar results are obt...

Abstract We prove that, for every graph F with at least one edge, there is a constant $$c_F$$ c F such that are graphs of arbitrarily large chromatic number and the same clique as in which -free induced subgraph has most . This generalises recent theorems Briański, Davi...

An induced matching in a graph is a set of edges such that no two edges in the set are joined by any third edge of the graph. An induced matching is maximum (MIM) if the number of edges in it is the largest among all possible induced matchings. It is known that finding the size of a MIM in a graph is NP-hard even if the graph is bipartite. It is also known that the size of a MIM in a chordal gr...

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⌋).

The matching forest problem in mixed graphs is a common generalization of the matching problem in undirected graphs and the branching problem in directed graphs. Giles presented an O(nm)-time algorithm for finding a maximum-weight matching forest, where n is the number of vertices and m is that of edges, and a linear system describing the matching forest polytope. Later, Schrijver proved total ...

Let α′ and μi denote the matching number of a non-empty simple graph G with n vertices ith smallest eigenvalue its Laplacian matrix, respectively. In this paper, we prove tight lower bound α′≥min⌈μ2μn(n−1)⌉,⌈12(n−1)⌉.This strengthens result Brouwer Haemers who proved that if is even 2μ2≥μn, then has perfect matching. A factor-critical for every vertex v∈V(G), G−v We also an analogue to mentione...