Quasi total double Roman domination stability in graphs

A quasi total double Roman dominating function (QTDRD-function) on a graph G=(V(G),E(G)) is a function f:V(G)→{0,1,2,3} having the property that (i) if f(v)=0, then the vertex v must have at least two neighbors assigned 2 under f or one neighbor w with f(w)=3, and if f(v)=1, then the vertex v must have at least one neighbor w with f(w)≥2, and (ii) if x is an isolated vertex in the subgraph of G induced by the set of vertices assigned nonzero values, then f(x)=2. The weight of a QTDRD-function f is the sum of its function values over the whole vertices, and the quasi total double Roman domination number γqtdR(G) equals the minimum weight of a QTDRD-function on G. The γqtdR-stability stγqtdR(G) (resp. γqtdR−-stability stγqtdR−(G), γqtdR+-stability stγqtdR+(G)) of G is defined as the order of the smallest set of vertices whose removal changes (resp. decreases, increases) the quasi total double Roman domination number. In this paper, we first determine the exact values of γqtdR-stability for some special classes of graphs, and then we present some bounds on stγqtdR(G) and stγqtdR−(G). We also characterize the graphs with large stγqtdR−(G). Moreover, we show that if T is a nontrivial tree, then stγqtdR(T)≤2, and if further T has maximum degree Δ≥3, then stγqtdR−(T)≤Δ−1.