On minimum cutsets in independent domination vertex-critical graphs

Abstract

© 2018, University of Queensland. All rights reserved. Let ? i (G) denote the independent domination number of G. A graph G is said to be k-? i -vertex-critical if ? i (G) = k and for each x ? V (G), ? i (G - x) < k. In this paper, we show that for any k-? i -vertex-critical graph H of order n with k = 3, there exists an n-connected k-? i -vertex-critical graph G H containing H as an induced subgraph. Consequently, there are infinitely many non-isomorphic connected k-? i -vertex-critical graphs. We also establish a number of properties of connected 3-? i -vertex-critical graphs. In particular, we derive an upper bound on ?(G-S), the number of components of G-S when G is a connected 3-? i -vertex-critical graph and S is a minimum cutset of G with |S| = 3.