Alimadadi, A., Eslahchi, C., Jafari Rad, N. (2012). A note on the total domination supercritical graphs. Transactions on Combinatorics, 1(3), 1-4. doi: 10.22108/toc.2012.1829
Abdollah Alimadadi; Changiz Eslahchi; Nader Jafari Rad. "A note on the total domination supercritical graphs". Transactions on Combinatorics, 1, 3, 2012, 1-4. doi: 10.22108/toc.2012.1829
Alimadadi, A., Eslahchi, C., Jafari Rad, N. (2012). 'A note on the total domination supercritical graphs', Transactions on Combinatorics, 1(3), pp. 1-4. doi: 10.22108/toc.2012.1829
Alimadadi, A., Eslahchi, C., Jafari Rad, N. A note on the total domination supercritical graphs. Transactions on Combinatorics, 2012; 1(3): 1-4. doi: 10.22108/toc.2012.1829
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 $\gamma_t(G+e)=k-2$ 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.
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
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