Outer Independent Double Roman Domination Stability in Graphs

An outer independent double Roman dominating function (OIDRDF) on a graph 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) the subgraph induced by the vertices assigned 0 under f is edgeless. The weight of an OIDRDF is the sum of its function values over all vertices, and the outer independent double Roman domination number γ o i d R ( G ) is the minimum weight of an OIDRDF on G . The γ o i d R -stability ( γ − o i d R -stability, γ + o i d R -stability) of G , denoted by s t γ o i d R ( G ) ( s t − γ o i d R ( G ) , s t + γ o i d R ( G ) ), is defined as the minimum size of a set of vertices whose removal changes (decreases, increases) the outer independent double Roman domination number. In this paper, we determine the exact values on the γ o i d R -stability of some special classes of graphs, and present some bounds on s t γ o i d R ( G ) . In addition, for a tree T with maximum degree Δ , we show that s t γ o i d R ( T ) = 1 and s t − γ o i d R ( T ) ≤ Δ , and characterize the trees that achieve the upper bound.