The t-latency bounded strong target set selection problem in some kinds of special family of graphs

Abstract

For a given simple graph $G=(V,E)$, a latency bound t and a threshold function $\theta (v)=\lceil \rho d(v)\rceil$, where $\rho \in (0,1)$ and d(v) denotes the degree of the vertex $v(\in V)$, a subset $S\subseteq V$ is called a strong target set if for each vertex $v\in S$, the number of its neighborhood in S not including itself is at least $\theta (v)$, and all vertices in V can be activated by S through a process with t rounds. Initially, all vertices in S become activated. At the ith round of the process, each vertex is activated if the number of active vertices in its neighbor after $i-1$ rounds exceeds its threshold. The $t$-Latency Bounded Strong Target Set Selection (t-LBSTSS) problem is to find such a strong target set S with the minimum cardinality in G. In general graphs, the t-LBSTSS problem is not only NP-hard, but also hard to be approximated. The aim of this paper is to find an optimal t-latency bounded strong target set for some special family of graphs. For a given simple graph G, a simple, tight but nontrivial inequality in terms of the number of edges in G is proposed to obtain the lower bound of the sum of degrees in a strong target set S to the t-LBSTSS problem. Moreover, a necessary and sufficient condition is presented for equality to hold. Finally, we give the exact formulas for the optimal solutions to the t-LBSTSS problem in two kinds of natural family of graphs, while it seems difficult to tell without the inequality given in this paper.