A set of vertices S is said to dominate the graph G if for each v /2 S, there is a vertex u 2 S with u adjacent to v, set S is called the dominating set in graph G. A dominating set, which induces a connected subgraph of G, is called connected dominating set. The smallest cardinality of any such dominating set (connected dominating set) is called the domination number of G and is denoted by (G)(c(G)). In this paper, we investigate graphs which are critical in the following sense: for each v, u 2 V (G) with v not adjacent to u, c(G + uv) < c(G). Thus G is k − c−critical if c(G) = k and for each edge e /2 E(G), we have c(G + e) < k.

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