Constructive characterizations concerning total outer-independent domination in subdivision trees
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2026
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| _version_ | 1866908909307953152 |
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| author | Cabrera-Martínez, A. López-Carmona, J. L. Serrano-Díaz, A. |
| author_facet | Cabrera-Martínez, A. López-Carmona, J. L. Serrano-Díaz, A. |
| contents | Let $G$ be a nontrivial connected graph with vertex set $V(G)$. A set of vertices $D\subseteq V(G)$ is called a total outer-independent dominating set of $G$ if every vertex of $G$ is adjacent to at least one vertex in $D$, and $V(G)\setminus D$ is an independent set of $G$. The total outer-independent domination number of $G$, denoted by $γ_t^{oi}(G)$, is the minimum cardinality among all total outer-independent dominating sets of $G$. The subdivision graph of $G$, denoted by $\mathtt{S}(G)$, is the graph obtained from $G$ by subdividing every edge exactly once. Cabrera-Martínez et al. [On the total outer-independent domination number of subdivision graphs, Comput. Appl. Math. 45 (2026) 315] proved that $\tfrac{4n(T)-l(T)-s(T)}{3}\leq γ_{t}^{oi}(\mathtt{S}(T))\leq \tfrac{4n(T)-l(T)+s(T)-2}{3}$ for any nontrivial tree $T$ of order $n(T)$ with $l(T)$ leaves and $s(T)$ support vertices. In this paper, we provide constructive characterizations of the families of trees that attain these bounds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_22884 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Constructive characterizations concerning total outer-independent domination in subdivision trees Cabrera-Martínez, A. López-Carmona, J. L. Serrano-Díaz, A. Combinatorics 05C05, 05C69, 05C75 Let $G$ be a nontrivial connected graph with vertex set $V(G)$. A set of vertices $D\subseteq V(G)$ is called a total outer-independent dominating set of $G$ if every vertex of $G$ is adjacent to at least one vertex in $D$, and $V(G)\setminus D$ is an independent set of $G$. The total outer-independent domination number of $G$, denoted by $γ_t^{oi}(G)$, is the minimum cardinality among all total outer-independent dominating sets of $G$. The subdivision graph of $G$, denoted by $\mathtt{S}(G)$, is the graph obtained from $G$ by subdividing every edge exactly once. Cabrera-Martínez et al. [On the total outer-independent domination number of subdivision graphs, Comput. Appl. Math. 45 (2026) 315] proved that $\tfrac{4n(T)-l(T)-s(T)}{3}\leq γ_{t}^{oi}(\mathtt{S}(T))\leq \tfrac{4n(T)-l(T)+s(T)-2}{3}$ for any nontrivial tree $T$ of order $n(T)$ with $l(T)$ leaves and $s(T)$ support vertices. In this paper, we provide constructive characterizations of the families of trees that attain these bounds. |
| title | Constructive characterizations concerning total outer-independent domination in subdivision trees |
| topic | Combinatorics 05C05, 05C69, 05C75 |
| url | https://arxiv.org/abs/2603.22884 |