Characterization of graphs with orientable total domination number equal to $|V|-1$
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866915008469794816 |
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| author | Blázsik, Zoltán L. Nagy, Leila Vivien |
| author_facet | Blázsik, Zoltán L. Nagy, Leila Vivien |
| contents | In a directed graph $D$, a vertex subset $S\subseteq V$ is a total dominating set if every vertex of $D$ has an in-neighbor from $S$. A total dominating set exists if and only if every vertex has at least one in-neighbor. We call the orientation of such directed graphs valid. The total domination number of $D$, denoted by $γ_t(D)$, is the size of the smallest total dominating set of $D$. For an undirected graph $G$, we investigate the upper (or lower) orientable total domination number of $G$, denoted by $\mathrm{DOM}_t(G)$ (or $\mathrm{dom}_t(G)$), that is the maximum (or minimum) of the total domination numbers over all valid orientations of $G$. We characterize those graphs for which $\mathrm{DOM}_t(G)=|V(G)|-1$, and consequently we show that there exists a family of graphs for which $\mathrm{DOM}_t(G)$ and $\mathrm{dom}_t(G)$ can be as far as possible, namely $\mathrm{DOM}_t(G)=|V(G)|-1$ and $\mathrm{dom}_t(G)=3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_04560 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characterization of graphs with orientable total domination number equal to $|V|-1$ Blázsik, Zoltán L. Nagy, Leila Vivien Combinatorics 05C69, 05C20 In a directed graph $D$, a vertex subset $S\subseteq V$ is a total dominating set if every vertex of $D$ has an in-neighbor from $S$. A total dominating set exists if and only if every vertex has at least one in-neighbor. We call the orientation of such directed graphs valid. The total domination number of $D$, denoted by $γ_t(D)$, is the size of the smallest total dominating set of $D$. For an undirected graph $G$, we investigate the upper (or lower) orientable total domination number of $G$, denoted by $\mathrm{DOM}_t(G)$ (or $\mathrm{dom}_t(G)$), that is the maximum (or minimum) of the total domination numbers over all valid orientations of $G$. We characterize those graphs for which $\mathrm{DOM}_t(G)=|V(G)|-1$, and consequently we show that there exists a family of graphs for which $\mathrm{DOM}_t(G)$ and $\mathrm{dom}_t(G)$ can be as far as possible, namely $\mathrm{DOM}_t(G)=|V(G)|-1$ and $\mathrm{dom}_t(G)=3$. |
| title | Characterization of graphs with orientable total domination number equal to $|V|-1$ |
| topic | Combinatorics 05C69, 05C20 |
| url | https://arxiv.org/abs/2411.04560 |