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 wit...