Outer independent signed double Roman domination

Abstract

Suppose $$[3]=\{0,1,2,3\}$$ and $$[3^{-}]=\{-1,1,2,3\}$$ . An outer independent signed double Roman dominating function (OISDRDF) of a graph $$\Gamma $$ is function $$l:V({\Gamma })\rightarrow [3^{-}]$$ for which (i) each vertex t with $$l(t)=-1$$ is joined to at least two vertices labeled a 2 or to at least one vertex z with $$l(z)=3$$ , (ii) each vertex t with $$l(t)=1$$ is joined to at least a vertex z with $$l(z)\ge 2,$$ (iii) $$l(N[t])=\sum _{w\in N[t]}l(w)\ge 1$$ occurs for each vertex t, (iv) the set of vertices labeled $$-1$$ under l is an independent set. The weight of an OISDRDF is the sum of its function values over all vertices, and the outer independent signed double Roman domination number (OISDRD-number) $$\gamma _{sdR}^{oi}(\Gamma )$$ is the minimum weight of an OISDRDF on $$\Gamma $$ . We first show that determining the number $$\gamma _{sdR}^{oi}(\Gamma )$$ is NP-complete for bipartite and chordal graphs. Then we provide exact values of this parameter for paths and cycles. Moreover, we show that for trees T of order $$n\ge 3,$$ $$\gamma _{sdR}^{oi}(\Gamma )\le n-1,$$ and we characterize extremal trees attaining this bound.