A note on the total domination supercritical graphs

‎Let G be a connected spanning subgraph of K_{s,s} and let H‎ ‎be the complement of G relative to K_{s,s}‎. ‎The graph G is‎ ‎k-supercritical relative to K_{s,s} if \gamma_t(G)=k‎ ‎and \gamma_t(G+e)=k-۲ for all e\in E(H)‎. ‎The ۲۰۰۲ paper by‎ ‎T.W‎. ‎Haynes‎, ‎M. A‎. ‎Henning and L.C‎. ‎van der Merwe‎, ‎``Total‎ ‎domination supercritical graphs with respect to relative‎ ‎complements‎" ‎that appeared in Discrete Mathematics‎, ‎۲۵۸ (۲۰۰۲)‎, ‎۳۶۱-۳۷۱‎, ‎presents a theorem (Theorem ۱۱) to produce (۲k‎ + ‎۲)-supercritical graphs relative to K_{۲k+۱‎, ‎۲k+۱} of diameter‎ ‎۵‎, ‎for each k\geq ۲‎. ‎However‎, ‎the families of graphs in their‎ ‎proof are not the case‎. ‎We present a correction of this theorem‎.

Keywords:

Total domination , Supercritical , Diameter