On minimum cutsets in independent domination vertex-critical graphs

Let gamma(i)(G) denote the independent domination number of G. A graph G is said to be k-gamma(i)-vertex-critical if gamma(i)(G) = k and for each x is an element of V (G), gamma(i)(G - x) < k. In this paper, we show that for any k-gamma(i)-vertex-critical graph H of order n with k >= 3, there exists an n-connected k-gamma(i)-vertexcritical graph G(H) containing H as an induced subgraph. Consequently, there are infinitely many non-isomorphic connected k-gamma(i)-vertex-critical graphs. We also establish a number of properties of connected 3-gamma(i)-vertex-critical graphs. In particular, we derive an upper bound on omega(G-S), the number of components of G-S when G is a connected 3-gamma(i)-vertex-critical graph and S is a minimum cutset of G with vertical bar S vertical bar >= 3.