Total Domination Supercritical Graphs With Respect to Relative Complements
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.
Haynes, Teresa W.; Henning, Michael A.; and Van Der Merwe, Lucas C.. 2002. Total Domination Supercritical Graphs With Respect to Relative Complements. Discrete Mathematics. Vol.258(1-3). https://doi.org/10.1016/S0012-365X(02)00537-X ISSN: 0012-365X