Double Italian domination in trees
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| Formato: | Preprint |
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2026
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| _version_ | 1866911552018317312 |
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| author | Shang, Weiping Zhang, Shanshan |
| author_facet | Shang, Weiping Zhang, Shanshan |
| contents | Let $G$ be a graph with vertex set $V=V(G)$. A double Roman dominating function on a graph $G$ is a function $f : V \to \{0,1,2,3\}$ satisfying the conditions that if $f(v) = 0$, then vertex $v$ must have at least two neighbors in $V_2$ or one neighbor in $V_3$, if $f(v) = 1$, then vertex $v$ must have at least one neighbor in $V_2 \cup V_3$. The weight of a double Roman dominating function $f$ is the sum $f(V) = \sum_{v \in V} f(v)$, and the double Roman domination number $γ_{dR}(G)$ is the minimum weight of a double Roman dominating function on $G$. A double Italian dominating function on a graph $G$ is a function $f : V \to \{0,1,2,3\}$ satisfying the condition that for every vertex $u \in V$, if $f(u) \in \{0,1\}$, then $\sum_{v \in N[u]} f(v) \ge 3$. The double Roman domination number $γ_{dI}(G)$ is the minimum weight of a double Italian dominating function on $G$. Mojdeh and Volkmann [D.A. Mojdeh and L. Volkmann, Roman {3}-domination (double Italian domination), Discrete Appl. Math. 283 (2020), 555--564] proved that $γ_{dI}(T) = γ_{dR}(T)$ for any tree $T$. However, we find that there is a minor issue in the proof. In this paper, we first prove that $γ_{dI}(T) \neq γ_{dR}(T)$. Subsequently, we present a sharp bound on the double Italian domination number of any non-trivial tree $T$, and characterize the trees attaining this bound. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_16438 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Double Italian domination in trees Shang, Weiping Zhang, Shanshan Combinatorics 05C70 Let $G$ be a graph with vertex set $V=V(G)$. A double Roman dominating function on a graph $G$ is a function $f : V \to \{0,1,2,3\}$ satisfying the conditions that if $f(v) = 0$, then vertex $v$ must have at least two neighbors in $V_2$ or one neighbor in $V_3$, if $f(v) = 1$, then vertex $v$ must have at least one neighbor in $V_2 \cup V_3$. The weight of a double Roman dominating function $f$ is the sum $f(V) = \sum_{v \in V} f(v)$, and the double Roman domination number $γ_{dR}(G)$ is the minimum weight of a double Roman dominating function on $G$. A double Italian dominating function on a graph $G$ is a function $f : V \to \{0,1,2,3\}$ satisfying the condition that for every vertex $u \in V$, if $f(u) \in \{0,1\}$, then $\sum_{v \in N[u]} f(v) \ge 3$. The double Roman domination number $γ_{dI}(G)$ is the minimum weight of a double Italian dominating function on $G$. Mojdeh and Volkmann [D.A. Mojdeh and L. Volkmann, Roman {3}-domination (double Italian domination), Discrete Appl. Math. 283 (2020), 555--564] proved that $γ_{dI}(T) = γ_{dR}(T)$ for any tree $T$. However, we find that there is a minor issue in the proof. In this paper, we first prove that $γ_{dI}(T) \neq γ_{dR}(T)$. Subsequently, we present a sharp bound on the double Italian domination number of any non-trivial tree $T$, and characterize the trees attaining this bound. |
| title | Double Italian domination in trees |
| topic | Combinatorics 05C70 |
| url | https://arxiv.org/abs/2603.16438 |