On the total Italian domination number in digraphs
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911108043898880 |
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| author | Dong, Changchang Guo, Yubao Lu, Mei Volkmann, Lutz |
| author_facet | Dong, Changchang Guo, Yubao Lu, Mei Volkmann, Lutz |
| contents | Consider a finite simple digraph $D$ with vertex set $V(D)$. An Italian dominating function (IDF) on $D$ is a function $f:V(D)\rightarrow\{0,1,2\}$ satisfying every vertex $u$ with $f(u)=0$ has an in-neighbor $v$ with $f(v)=2$ or two in-neighbors $w$ and $z$ with $f(w)=f(z)=1$. A total Italian dominating function (TIDF) on $D$ is an IDF $f$ such that the subdigraph $D[\{ u\, |\, f(u)\ge 1\}]$ contains no isolated vertices. The weight $ω(f)$ of a TIDF $f$ on $D$ is $\sum_{u\in V(D)}f(u)$. The total Italian domination number of $D$ is $γ_{tI}(D)=\min\{ ω(f)\, |\, \mbox{$f$ is a TIDF on $D$}\}$.
In this paper, we present bounds on $γ_{tI}(D)$, and investigate the relationship between several different domination parameters. In particular, we give the total Italian domination number of the Cartesian products $P_2\Box P_n$ and $P_3\Box P_n$, where $P_n$ represents a dipath with $n$ vertices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_17368 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the total Italian domination number in digraphs Dong, Changchang Guo, Yubao Lu, Mei Volkmann, Lutz Combinatorics Consider a finite simple digraph $D$ with vertex set $V(D)$. An Italian dominating function (IDF) on $D$ is a function $f:V(D)\rightarrow\{0,1,2\}$ satisfying every vertex $u$ with $f(u)=0$ has an in-neighbor $v$ with $f(v)=2$ or two in-neighbors $w$ and $z$ with $f(w)=f(z)=1$. A total Italian dominating function (TIDF) on $D$ is an IDF $f$ such that the subdigraph $D[\{ u\, |\, f(u)\ge 1\}]$ contains no isolated vertices. The weight $ω(f)$ of a TIDF $f$ on $D$ is $\sum_{u\in V(D)}f(u)$. The total Italian domination number of $D$ is $γ_{tI}(D)=\min\{ ω(f)\, |\, \mbox{$f$ is a TIDF on $D$}\}$. In this paper, we present bounds on $γ_{tI}(D)$, and investigate the relationship between several different domination parameters. In particular, we give the total Italian domination number of the Cartesian products $P_2\Box P_n$ and $P_3\Box P_n$, where $P_n$ represents a dipath with $n$ vertices. |
| title | On the total Italian domination number in digraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2406.17368 |