Sharp bounds for product and sum throttling numbers

TL;DR

The paper addresses the problem of establishing sharp, order-based bounds for throttling numbers in graph processes—including power domination, PSD forcing, and standard zero forcing—by introducing a new technique to bound initial-cost product throttling and to bound sum throttling for power domination. The authors derive tight results such as $\operatorname{th}_{\gamma_P}^{\times}(G) \le \frac{6n}{7}$ for connected graphs and $\operatorname{th}_{\gamma_P}(G) \le \left\lfloor \frac{n}{3} \right\rfloor + 2$, with proofs leveraging edge-maximum minimum dominating sets and external private neighbors, plus corollaries linking equality to the domination number. They also establish sharp, general bounds on how product throttling numbers change under fundamental graph operations (edge/vertex deletion, contraction, subdivision) across the throttling variants, supported by extensive sharpness examples. These results illuminate the intrinsic limits and stability of throttling parameters under graph edits, with implications for network monitoring and related combinatorial processes.

Abstract

Throttling in graphs optimizes a sum or product of resources used, such as the number of vertices in an initial set, and time required, such as the propagation time, to complete a given task. We introduce a new technique to establish sharp upper bounds in terms of graph order for sum throttling and initial cost product throttling for power domination. Furthermore, we establish sharp bounds on possible changes of the product throttling number, both with and without initial cost, caused by certain graph operations for standard zero forcing, positive semidefinite forcing, and power domination.

Figures (7)

• Figure 2.1: A graph $G$ that achieves the upper bound in Theorem \ref{['thm:6/7']}.
• Figure 3.1: Adding the dotted edge $e$ to the spider $S(3,3,3,3,3,3)$ doubles $\operatorname{th}_Y^\ast$ and $\operatorname{th}_Y^\times$ for $Y\in \{\gamma_P,\operatorname{Z}_+\}$.
• Figure 3.2: Each graph $H_i$ includes the dotted edge $e$. Deleting $e$ doubles the listed parameter(s): (a) $\operatorname{th}_{\gamma_P}^\ast$; (b) $\operatorname{th}_{\gamma_P}^\times$; (c) $\operatorname{th}_+^\ast$, $\operatorname{th}_+^\times$.
• Figure 3.3: Graph $G$ is constructed from spider $S(2,2,2,2)$ by adding an independent twin $x$ of the center vertex $c$, which doubles $\operatorname{th}_Y^\ast$ and $\operatorname{th}_Y^\times$ for $Y\in\{\gamma_P,\operatorname{Z}_+\}$.
• Figure 3.4: (a) shows graph $G$ with edge $e$, and (b) shows $G/e$. Contracting $e$ halves $\operatorname{th}_{\gamma_P}^\ast$, $\operatorname{th}_+^\ast$, and $\operatorname{th}_{\gamma_P}^\times$.

Theorems & Definitions (30)

• Remark 1.1
• Theorem 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Example 2.4
• Theorem 2.5
• proof
• Corollary 2.6