Partially critical indecomposable graphs

Given a graph G = (V, E), with each subset X of V is associated the subgraph G(X) of G induced by X. A subset I of V is an interval of G provided that for any a, b ∈ I and x ∈ V \ I , {a, x} ∈ E if and only if {b, x} ∈ E. For example, ∅, {x}, where x ∈ V , and V are intervals of G called trivial intervals. A graph is indecomposable if all its intervals are trivial; otherwise, it is decomposable. Given an indecomposable graph G = (V, E), consider a proper subset X of V such that |X| ≥ 4 and G(X) is indecomposable. The graph G is critical according to G(X) if for every x ∈ V \ X, G(V \ {x}) is decomposable. A graph is partially critical if it is critical according to one of its indecomposable subgraphs containing at least 4 vertices. In this paper, we characterize the partially critical graphs. Mathematics Subject Classifications (1991): 05C75.