On the Existence of K-Partite or KP-Free Total Domination Edge-Critical Graphs
A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γt(G). The graph G is 3t-critical if γt(G)=3 and γt(G+e)=2 for every edge e in the complement of G. We show that no bipartite graph is 3t-critical. The tripartite 3 t-critical graphs are characterized. For every k<3, we prove that there are only a finite number of 3t-critical k-partite graphs. We show that the 5-cycle is the only 3t-critical K3-free graph and that there are only a finite number of 3t-critical K4-free graphs.
Haynes, Teresa W.; Henning, Michael A.; Van Der Merwe, Lucas C.; and Yeo, Anders. 2011. On the Existence of K-Partite or KP-Free Total Domination Edge-Critical Graphs. Discrete Mathematics. Vol.311(13). 1142-1149. https://doi.org/10.1016/j.disc.2010.07.018 ISSN: 0012-365X