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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References

A. Alimadadi, Ch. Eslahchi, T. W. Haynes, M. A. Henning, N. Jafari Rad and L. C. van der Merwe (2012). Erratum to "Total domination supercritical graphs with respect to relative complements. [Discrete Math., {bf 258} (2002) 361-371], Discrete Math. 312, 1076
T. W. Haynes, S. T. Hedetniemi and P. J. Slater (1998). Fundamentals of Domination in Graphs. Marcel Dekker, NewYork.
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
Transactions on Combinatorics

Volume 1, Issue 3
September 2012
Pages 1-4

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