Some toughness results in independent domination critical graphs

A subset S of V (G) is an independent dominating set of G if S is independent and each vertex of G is either in S or adjacent to some vertex of S. Let i(G) denote the minimum cardinality of an independent dominating set of G. A graph G is k-i-critical if i(G) = k, but i(G+uv) < k for any pair of non-adjacent vertices u and v of G. In this paper, we establish that if G is a connected 3-i-critical graph and S is a vertex cutset of G with |S| ≥ 3, then ω(G−S) ≤ 1+ √ 8|S|+1 2 , improving a result proved by Ao [3], where ω(G−S) denotes the number of components of G−S. We also provide a characterization of the connected 3-i-critical graphs G attaining the maximum number of ω(G− S) when S is a minimum cutset of size 2 or 3.