A note on Cops and Robbers, independence number, domination number and diameter

TL;DR

This paper explores relationships between key graph invariants such as diameter, domination number, independence number, and cop number, providing new bounds that connect these parameters in connected graphs.

Contribution

It introduces novel bounds on the cop number based on diameter, domination, and independence numbers, advancing understanding of graph pursuit-evasion dynamics.

Findings

Bound c(G) by independence number and diameter.

Bound c(G) by domination number and diameter.

Provides inequalities relating graph parameters.

Abstract

We study relations between diameter $D(G)$, domination number $\gamma(G)$, independence number $\alpha(G)$ and cop number $c(G)$ of a connected graph $G$, showing (i.) $c(G) \leq \alpha(G)-\lfloor \frac{D(G)-3}{2} \rfloor$, and (ii.) $c(G) \leq \gamma (G) - \frac{D(G)}{3} + O (\sqrt{D(G)})$.