Approximation algorithm for the minimum partial connected Roman dominating set problem

Given a graph \(G=(V,E)\) and a function \(r:V\mapsto \{0,1,2\}\), a node \(v\in V\) is said to be Roman dominated if \(r(v)=1\) or there exists a node \(u\in N_G[v]\) such that \(r(u)=2\), where \( N_G[v]\) is the closed neighbor set of v in G. For \(i\in \{0,1,2\}\), denote \(V_r^i\) as the set of nodes with value i under function r. The cost of r is defined to be \(c(r)=|V_r^1|+2|V_r^2|\). Given a positive integer \(Q\le |V|\), the minimum partial connected Roman dominating set (MinPCRDS) problem is to compute a minimum cost function r such that at least Q nodes in G are Roman dominated and the subgraph of G induced by \(V_r^1\cup V_r^2\) is connected. In this paper, we give a \((3\ln |V|+9)\)-approximation algorithm for the MinPCRDS problem.