A note on the total domination supercritical graphs

Abstract

‎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 ‎<em>$k$-supercritical</em> relative to $K_{s,s}$ if $gamma_t(G)=k$‎ ‎and $gamma_t(G+e)=k-2$ for all $ein E(H)$‎. ‎The 2002 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, ‎258 (2002)‎, ‎361-371‎, ‎presents a theorem (Theorem 11) to produce $(2k‎ + ‎2)$-supercritical graphs relative to $K_{2k+1‎, ‎2k+1}$ of diameter‎ ‎$5$‎, ‎for each $kgeq 2$‎. ‎However‎, ‎the families of graphs in their‎ ‎proof are not the case‎. ‎We present a correction of this theorem‎.