On minimum cutsets in independent domination vertexcritical graphs
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© 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 vertexcritical if ? i (G) = k and for each x ? V (G), ? i (G  x) < k. In this paper, we show that for any k? i vertexcritical graph H of order n with k = 3, there exists an nconnected k? i vertexcritical graph G H containing H as an induced subgraph. Consequently, there are infinitely many nonisomorphic connected k? i vertexcritical graphs. We also establish a number of properties of connected 3? i vertexcritical graphs. In particular, we derive an upper bound on ?(GS), the number of components of GS when G is a connected 3? i vertexcritical graph and S is a minimum cutset of G with S = 3.
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