A note on the total domination supercritical graphs

Document Type: Research Paper

Authors

1 Shahid Beheshti University

2 Shahrood University of Technology

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 ‎$k$-supercritical relative to $K_{s,s}$ if $\gamma_t(G)=k$‎ ‎and $\gamma_t(G+e)=k-2$ for all $e\in E(H)$‎. ‎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 $k\geq 2$‎. ‎However‎, ‎the families of graphs in their‎ ‎proof are not the case‎. ‎We present a correction of this theorem‎.

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T. W. Haynes, M. A. Henning, and L. C. van der Merwe (2002). Total domination supercritical graphs with respect to relative complements. Total domination supercritical graphs with respect to relative complements. 258, 361-371