A maximum stable set in a graph G is a stable set of maximum size. S is a local maximum stable set of G, and we write S ∈ Ψ(G), if S is a maximum stable set of the subgraph spanned by S∪N(S), where N(S) is the neighborhood of S. A matching M is uniquely restricted if its saturated vertices induce a subgraph which has a unique perfect matching, namely M itself. Nemhauser and Trotter Jr. [12], pr...

For a graph property P, we define a P-matching as a set M of disjoint edges such that the subgraph induced by the vertices incident to M has property P. Previous examples include strong/induced matchings and uniquely restricted matchings. We explore the general properties of P-matchings, but especially the cases where P is the property of being acyclic or the property of being disconnected. We ...

Let G be a cubic graph with a perfect matching. An edge e of G is a forcing edge if it is contained in a unique perfect matching M , and the perfect matching M is called uniquely forced. In this paper, we show that a 3-connected cubic graphwith a uniquely forced perfect matching is generated from K4 via Y → 1-operations, i.e., replacing a vertex by a triangle, and further a cubic graph with a u...

A maximum stable set in a graph G is a stable set of maximum size. S is a local maximum stable set of G, and we write S ∈0(G), if S is a maximum stable set of the subgraph spanned by S ∪ N (S), where N (S) is the neighborhood of S. A matching M is uniquely restricted if its saturated vertices induce a subgraph which has a unique perfect matching, namely M itself. Nemhauser and Trotter Jr. (Math...