On positive-influence target-domination

Consider a graph $$G=(V,E)$$G=(V,E) and a vertex subset $$A \subseteq V$$A⊆V. A vertex v is positive-influence dominated by A if either v is in A or at least half the number of neighbors of v belong to A. For a target vertex subset $$S \subseteq V$$S⊆V, a vertex subset A is a positive-influence target-dominating set for target set S if every vertex in S is positive-influence dominated by A. Given a graph G and a target vertex subset S, the positive-influence target-dominating set (PITD) problem is to find the minimum positive-influence dominating set for target S. In this paper, we show two results: (1) The PITD problem has a polynomial-time $$(1 + \log \lceil \frac{3}{2} \Delta \rceil )$$(1+log⌈32Δ⌉)-approximation in general graphs where $$\Delta $$Δ is the maximum vertex-degree of the input graph. (2) For target set S with $$|S|=\Omega (|V|)$$|S|=Ω(|V|), the PITD problem has a polynomial-time O(1)-approximation in power-law graphs.