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Total double Roman domination stability in graphs | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 17، دوره 11، شماره 1، خرداد 2026، صفحه 247-258 اصل مقاله (386.67 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2025.30402.2453 | ||
| نویسندگان | ||
| Ziqiang Xu1؛ Saeed Kosari* 2؛ Mina Esmaeili3؛ Aysha Khan4؛ Lutz Volkmann5 | ||
| 1Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China | ||
| 2Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China | ||
| 3Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, I.R. Iran | ||
| 4University of Technology and Applied Sciences, Musannah, Oman | ||
| 5RWTH Aachen, 52056 Aachen, Germany | ||
| چکیده | ||
| Let $G$ be a graph with vertex set $V(G)$. A total double Roman dominating function (TDRD-function) on a graph $G$ with no isolated vertices is a function $f :V(G)\to \{0, 1, 2, 3\}$ satisfying the conditions: $(i)$ if $f(v)=0$, then the vertex $v$ must be adjacent to at least two vertices assigned $2$ or one vertex assigned $3$ under $f$, and if $f(v)=1$, then the vertex $v$ must be adjacent to at least one vertex assigned $2$ or $3$ and $(ii)$ the subgraph of $G$ induced by the set $\{v \in V(G) \mid f(v)\neq 0\}$ has no isolated vertices. The weight of a TDRD-function $f$ is the sum of its function values over all vertices, and the minimum weight of a TDRD-function on $G$ is the total double Roman domination number, $\gamma_{tdR}(G)$. The $\gamma_{tdR}$-stability ($\gamma^-_{tdR}$-stability, $\gamma^+_{tdR}$-stability) of $G$, denoted by ${\rm st}_{\gamma_{tdR}}(G)$ (resp. ${\rm st}^-_{\gamma_{tdR}}(G)$, ${\rm st}^+_{\gamma_{tdR}}(G)$), is defined as the minimum size of a set of vertices whose removal changes (resp. decreases, increases) the total double Roman domination number. In this paper, we first determine the exact values of the $\gamma_{tdR}$-stability of some special classes of graphs, and then we present some bounds on ${\rm st}_{\gamma_{tdR}}(G)$, ${\rm st}^-{\gamma_{tdR}}(G)$ and ${\rm st}^+_{\gamma_{tdR}}(G)$). In particular, for a graph $G$ with maximum degree $\Delta\ge 3$, we show that ${\rm st}^-_{\gamma_{tdR}}(G)\leq \Delta-1$. | ||
| کلیدواژهها | ||
| total double Roman domination؛ total double Roman domination stability | ||
| مراجع | ||
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