Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 11 1 2026 03 01 Total double Roman domination stability in graphs 247 258 14966 10.22049/cco.2025.30402.2453 EN Ziqiang Xu Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China Saeed Kosari Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China 0000-0002-1427-5473 Mina Esmaeili Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, I.R. Iran Aysha Khan University of Technology and Applied Sciences, Musannah, Oman Lutz Volkmann RWTH Aachen, 52056 Aachen, Germany Journal Article 2025 03 07 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 ${\rm st}^-_{\gamma_{tdR}}(G)\leq \Delta-1$. https://comb-opt.azaruniv.ac.ir/article_14966_46b3bba02f971dc5ccfed81a69b04ac3.pdf