total double roman domination stability in graphs

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 $deltage 3$, we show that ${rm st}^-_{gamma_{tdr}}(g)leq delta-1$.