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 (a ₁,b ₁), ..., (a _k ,b _k ) such that induced subgraphs on subset $V-\{a_{j_1},b_{j_1}, \ldots, a_{j_s},b_{j_s}\}$ are also X-critical for any collection of pairs $\{(a_{j_1},b_{j_1}), \ldots, (a_{j_s},b_{j_s})\}$.