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.

对于给定的简单图\（G =（V，E）\），等待时间限制为t和阈值函数\（\ theta（v）= \ lceil \ rho d（v）\ rceil \），其中\（\ rho \ in（0,1）\）和d（v）表示顶点\（v（\ in V）\）的度数，子集\（S \ subseteq V \）被称为强目标集，如果每个顶点\（v \ in S \）中，其在S中的邻域数量（不包括自身）至少为\（\ theta（v）\），并且V中的所有顶点都可以由S通过t的过程来激活回合。最初，S中的所有顶点都被激活。在该过程的第i轮中，如果\（i-1 \）轮之后其邻居中的活动顶点数超过其阈值，则会激活每个顶点。的\（T \） -Latency界很好目标集选择（吨-LBSTSS）问题是找到这样的强目标组小号与最小基数ģ。在一般图中，t -LBSTSS问题不仅是NP难题，而且也难以近似。本文的目的是为某些特殊的图族找到最优的t延迟有界强目标集。对于给定的简单图G中，在边缘的数量方面具有简单，紧但非平凡不等式ģ提出以获得较低一个强有力的目标组结合度的总和的小号到吨-LBSTSS问题。此外，提出了保持平等的必要和充分条件。最后，我们给出了两种自然图族中t -LBSTSS问题的最优解的精确公式，但是如果没有本文给出的不等式，似乎很难分辨。