Conditionally critical indecomposable graphs

Abstract

Let X be a subset of vertices of an undirected graph G = (V, E). G is X-critical if it is indecomposable and its induced subgraph on X vertices is also indecomposable, but all induced subgraphs on V - {w} are decomposable for all w ∈ V - X. We present two results in this paper. The first result states that if G is X-critical, then for every w ∈ V - {x}, G[V -{w}] has a unique non-trivial module and its cardinality is either 2 or |V| - 2. The second result is that the vertices of V - X can be paired up as (a1, b1), . . . , (ak, bk) such that induced subgraphs on subset V - {aj1, bj1, . . . , ajs, bjs} are also X-critical for any collection of pairs {(a j1, bj1), . . . ,(ajs, bjs)}. © Springer-Verlag Berlin Heidelberg 2005.