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Total Domination Supercritical Graphs With Respect to Relative ComplementsHaynes, Teresa W., Henning, Michael A., Van Der Merwe, Lucas C. 06 December 2002 (has links)
A set S of vertices of a graph G is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number γt(G) is the minimum cardinality of a total dominating set of G. Let G be a connected spanning subgraph of Ks,s, and let H be the complement of G relative to Ks,s; that is, Ks,s, = G ⊕ H is a factorization of Ks,s. The graph G is k-supercritical relative to Ks,s, if γt(G) = k and γ1(G + e) = k - 2 for all e ∈ E(H). Properties of k-supercritical graphs are presented, and k-supercritical graphs are characterized for small k.
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