On Tight Components and Anti-Tight Components

A graph G = (V, E) is called factor-critical if G = ∅ and G − v has a perfect matching for every vertex v ∈ V (G). A factor-critical graph G is tight (anti-tight, respectively) if for any v ∈ V (G), any perfect matching M in G − v, and any e ∈ M , |N (v) ∩ V (e)| = 1 (|N (v) ∩ V (e)| = 2, respectively), where N (v) denotes the neighborhood of v and V (e) denotes the set of vertices incident with e. A graph G is minimally anti-tight if G is anti-tight but G − e is not anti-tight for every e ∈ E(G). In this paper, we prove that a connected graph is tight iff every block of the graph is an odd clique, and that every minimally anti-tight graph is triangle-free.