Domination number of modular product graphs
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2024
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| _version_ | 1866917629626679296 |
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| author | Bermudo, Sergio Peterin, Iztok Sedlar, Jelena Škrekovski, Riste |
| author_facet | Bermudo, Sergio Peterin, Iztok Sedlar, Jelena Škrekovski, Riste |
| contents | The modular product $G\diamond H$ of graphs $G$ and $H$ is a graph on vertex set $V(G)\times V(H)$. Two vertices $(g,h)$ and $(g^{\prime},h^{\prime})$ of $G\diamond H$ are adjacent if $g=g^{\prime}$ and $hh^{\prime}\in E(H)$, or $gg^{\prime}\in E(G)$ and $h=h^{\prime}$, or $gg^{\prime}\in E(G)$ and $hh^{\prime}\in E(H)$, or (for $g\neq g^{\prime}$ and $h\neq h^{\prime}$) $gg^{\prime}\notin E(G)$ and $hh^{\prime}\notin E(H)$. A set $D\subseteq V(G)$ is a dominating set of $G$ if every vertex outside of $D$ contains a neighbor in $D$. A set $D\subseteq V(G)$ is a total dominating set of $G$ if every vertex of $G$ contains a neighbor in $D$. The domination number $γ(G)$ (resp. total domination number $γ_{t}(G)$) of $G$ is the minimum cardinality of a dominating set (resp. total dominating set) of $G$. In this work we give several upper and lower bounds for $γ(G\diamond H)$ in terms of $γ(G),$ $γ(H)$, $γ_{t}(\overline{G})$ and $γ_{t}(\overline{H})$, where $\overline{G}$ is the complement graph of $G$. Further, we fully describe graphs where $γ(G\diamond H)=k$ for $k\in\{1,2,3\}$. Several conditions on $G$ and $H$ under which $γ(G\diamond H)$ is at most $4$ and $5$ are also given. A new type of simultaneous domination $\barγ(G)$, defined as the smallest number of vertices that dominates $G$ and totally dominates the complement of $G,$ emerged as useful and we believe it could be of independent interest. We conclude the paper by proposing few directions for possible further research. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_02853 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Domination number of modular product graphs Bermudo, Sergio Peterin, Iztok Sedlar, Jelena Škrekovski, Riste Combinatorics 05C69, 05C76 The modular product $G\diamond H$ of graphs $G$ and $H$ is a graph on vertex set $V(G)\times V(H)$. Two vertices $(g,h)$ and $(g^{\prime},h^{\prime})$ of $G\diamond H$ are adjacent if $g=g^{\prime}$ and $hh^{\prime}\in E(H)$, or $gg^{\prime}\in E(G)$ and $h=h^{\prime}$, or $gg^{\prime}\in E(G)$ and $hh^{\prime}\in E(H)$, or (for $g\neq g^{\prime}$ and $h\neq h^{\prime}$) $gg^{\prime}\notin E(G)$ and $hh^{\prime}\notin E(H)$. A set $D\subseteq V(G)$ is a dominating set of $G$ if every vertex outside of $D$ contains a neighbor in $D$. A set $D\subseteq V(G)$ is a total dominating set of $G$ if every vertex of $G$ contains a neighbor in $D$. The domination number $γ(G)$ (resp. total domination number $γ_{t}(G)$) of $G$ is the minimum cardinality of a dominating set (resp. total dominating set) of $G$. In this work we give several upper and lower bounds for $γ(G\diamond H)$ in terms of $γ(G),$ $γ(H)$, $γ_{t}(\overline{G})$ and $γ_{t}(\overline{H})$, where $\overline{G}$ is the complement graph of $G$. Further, we fully describe graphs where $γ(G\diamond H)=k$ for $k\in\{1,2,3\}$. Several conditions on $G$ and $H$ under which $γ(G\diamond H)$ is at most $4$ and $5$ are also given. A new type of simultaneous domination $\barγ(G)$, defined as the smallest number of vertices that dominates $G$ and totally dominates the complement of $G,$ emerged as useful and we believe it could be of independent interest. We conclude the paper by proposing few directions for possible further research. |
| title | Domination number of modular product graphs |
| topic | Combinatorics 05C69, 05C76 |
| url | https://arxiv.org/abs/2404.02853 |