Summary
A dominating set of vertices S of a graph G is connected if the subgraph G[S] is connected. Let γ c (G) denote the size of any smallest connected dominating set in G. Graph G is k-γ-connected-vertex-critical (abbreviated “kcvc”) if \(\gamma_{c}(G)=\nobreak k\) , but if any vertex v is deleted from G, then γ c (G−v)≤k−1.
This concept of vertex criticality stands in contrast to the concept of criticality with respect to edge addition in which a graph G is defined to be k-connected-critical if the connected domination number of G is k, but if any edge is added to G, the connected domination number falls to k−1.
It is well-known that the only 1cvc graph is K 1 and the 2cvc graphs are obtained from the even complete graphs K 2n , with n≥2, by deleting a perfect matching. In this paper we survey some recent results for the case when γ c =3. In Sect. 14.2 we present some recently derived basic properties of 3cvc graphs, especially with respect to connectivity, and then present three new infinite families of 3cvc graphs. In Sect. 14.3, we present some new matching results for 3cvc graphs.
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Plummer, M.D. (2009). On the Structure of Graphs Vertex Critical with Respect to Connected Domination. In: Cook, W., Lovász, L., Vygen, J. (eds) Research Trends in Combinatorial Optimization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-76796-1_14
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