The structure of 4--critical graphs with a cut vertex

Let γ ( G ) denote the minimum cardinality of a dominating set for G . A graph G is said to be k - γ -critical if γ ( G ) = k , but γ ( G + e ) < k for each edge e ∈ E ( G ¯ ) . In this paper, we provide the structure of 4- γ -critical connected graphs with a cut vertex. We establish that such graphs of even order contain a perfect matching. This result partially resolves a problem posed by Sumner and Wojcicka in 1998. They asked whether every k - γ -critical graph of even order contains a perfect matching for k ≥ 4 .