Total Domination Good Vertices in 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 total domination number γt(G) is the minimum cardinality of a total dominating set of G. A vertex that is contained in some minimum total dominating set of a graph G is a good vertex, otherwise it is a bad vertex. We determine for which triples (x, y, z) there exists a connected graph G with γt(G) = x and with y good vertices and z bad vertices, and we give graphs realizing these triples.
Haynes, Teresa W.; and Henning, Michael A.. 2002. Total Domination Good Vertices in Graphs. Australasian Journal of Combinatorics. Vol.26 305-315. https://ajc.maths.uq.edu.au/pdf/26/ajc_v26_p305.pdf ISSN: 1034-4942