Geodesic
Some Toughness Results in Independent Domination Critical Graphs
Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 4, pp. 703-713

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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) lt; 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.
Keywords: domination critical, toughness 
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     author = {Ananchuen, Nawarat and Ananchuen, Watcharaphong},
     title = {Some {Toughness} {Results} in {Independent} {Domination} {Critical} {Graphs}},
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Ananchuen, Nawarat; Ananchuen, Watcharaphong. Some Toughness Results in Independent Domination Critical Graphs. Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 4, pp. 703-713. http://geodesic.mathdoc.fr/item/DMGT_2015_35_4_a8/