[Paper Review] Throttling numbers for adversaries on connected graphs

Jesse Geneson|arXiv (Cornell University)|Jun 17, 2019

Advanced Graph Theory Research13 references4 citations

TL;DR

This paper resolves two open problems on throttling numbers in cop-and-robber games on graphs by proving a tighter upper bound of $\frac{\sqrt{14}}{2}\sqrt{n} + O(1)$ for the throttling number of trees, improving upon the prior $2\sqrt{n}$ bound. It also establishes an $O(\sqrt{n})$ bound for cactus graphs and presents a new lower bound of $1.4502\sqrt{n}$ for certain trees, demonstrating that the $2\sqrt{n}$ conjecture cannot hold asymptotically.

ABSTRACT

In this paper, we answer two open problems from [Breen et al., Throttling for the game of Cops and Robbers on graphs, Discrete Math., 341 (2018) 2418-2430]. The throttling number $th_c(G)$ of a graph $G$ is the minimum possible value of $k + capt_k(G)$ over all positive integers $k$, where $capt_k(G)$ is the number of rounds needed for $k$ cops to capture the robber on $G$. One of the problems from [Breen et al., 2018] was to determine whether there exists a family of trees $T$ of order $n$ for which $th_c(T)$ is asymptotically equal to $2 \sqrt{n}$. We show that such a family cannot exist by improving the upper bound on $\displaystyle \max_{T} th_c(T)$ for all trees $T$ of order $n$ from $2 \sqrt{n}$ to $\frac{\sqrt{14}}{2} \sqrt{n} + O(1)$. We prove this bound by deriving a more general throttling bound for connected graphs that applies to multiple graph adversaries, including the robber and the gambler. This also improves the best known upper bounds on $th_c(G)$ for chordal graphs and unicyclic graphs $G$, as well as throttling numbers for positive semidefinite (PSD) zero forcing on trees. In addition to the results about cop versus robber, we use our general throttling bound to improve previous upper bounds on throttling numbers for the cop versus gambler game on connected graphs. Another open problem from [Breen et al., 2018] was to obtain a bound on $th_c(G)$ for cactus graphs $G$. We prove an $O(\sqrt{n})$ bound for all cactus graphs $G$ of order $n$. Furthermore, we exhibit a family of trees $T$ of order $n$ that have $th_c(T) > 1.4502 \sqrt{n}$ for all $n$ sufficiently large, improving on the previous lower bound of $\lceil \sqrt{2n}-\frac{1}{2} ceil + 1$ on $\displaystyle \max_{T} th_c(T)$ for trees $T$ of order $n$.

Motivation & Objective

• To resolve an open problem on whether any family of trees has throttling number asymptotically equal to $2\sqrt{n}$.
• To establish tighter upper bounds on the throttling number for trees, chordal graphs, unicyclic graphs, and cactus graphs.
• To improve the known lower bound on the maximum throttling number for trees of order $n$.
• To extend the throttling framework to multiple adversaries, including the gambler and the robber, via a unified analytical method.

Proposed method

• Derives a general throttling bound applicable to multiple graph adversaries, including the cop-and-robber and cop-and-gambler games.
• Uses a geometric partitioning strategy by cutting intervals of length $\lfloor r\sqrt{n}\rfloor$ from spider legs to bound capture time and number of cops.
• Applies optimization techniques to minimize $k + \operatorname{capt}_k(G)$ by balancing cop count and capture time across graph structures.
• Constructs unbalanced spiders with $\lfloor a\sqrt{n}\rfloor$ short legs of length $\lfloor c\sqrt{n}\rfloor$ and one long leg to derive lower bounds.
• Employs case analysis based on whether short legs are initially occupied by cops or not to bound the throttling number.
• Leverages known equivalences between cop-and-robber throttling and positive semidefinite (PSD) zero forcing throttling on trees.

Experimental results

Research questions

• RQ1 Is there a family of trees for which the throttling number $\operatorname{th_c}(T)$ is asymptotically $2\sqrt{n}$?
• RQ2 Can the throttling number of cactus graphs be bounded by $O(\sqrt{n})$?
• RQ3 What is the best possible lower bound for $\max_T \operatorname{th_c}(T)$ over all trees $T$ of order $n$?
• RQ4 Can a unified throttling bound be derived that applies to multiple adversaries, such as the robber and the gambler?
• RQ5 What is the tightest upper bound for the throttling number of spiders with arbitrary leg lengths?

Key findings

• The throttling number for any tree $T$ of order $n$ is at most $\frac{\sqrt{14}}{2}\sqrt{n} + O(1)$, improving the prior upper bound of $2\sqrt{n}$.
• For all cactus graphs $G$ of order $n$, $\operatorname{th_c}(G) = O(\sqrt{n})$, resolving an open problem from Breen et al. (2018).
• A family of unbalanced spiders $S_{n,a,c}$ achieves $\operatorname{th_c}(S_{n,a,c}) > 1.4502\sqrt{n}$ for all sufficiently large $n$, improving the prior lower bound of $\left\lceil \sqrt{2n} - \frac{1}{2} \right\rceil + 1$.
• For any spider $S$ of order $n$, $\operatorname{th_c}(S) \leq \sqrt{3}\sqrt{n} + O(1)$, which holds for all spiders, including those with legs of unequal length.
• The general throttling bound technique applies beyond the robber to other adversaries, such as the gambler, improving known upper bounds on expected capture time.
• The results extend to positive semidefinite (PSD) zero forcing throttling on trees, confirming that $\operatorname{th_+}(T) \leq \frac{\sqrt{14}}{2}\sqrt{n} + O(1)$ for all trees $T$.