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 {it $k$-supercritical} 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 Me...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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University of Isfahan
2012-09-01
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| Series: | Transactions on Combinatorics |
| Subjects: | |
| Online Access: | http://www.combinatorics.ir/?_action=showPDF&article=1829&_ob=233087645ace589c1ff0904792d8fee7&fileName=full_text.pdf |
| Summary: | 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 {it $k$-supercritical} 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. |
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| ISSN: | 2251-8657 2251-8665 |