Extremal Cat Herding

TL;DR

This paper studies a graph game called Cat Herding, analyzing conditions for the cat's capture or evasion, classifying graphs by their cat number, and exploring infinite cases and future research directions.

Contribution

It introduces a reduction construction preserving cat number, classifies graphs with cat number ≤3, and characterizes infinite graphs where the cat can evade capture indefinitely.

Findings

Classified all graphs with cat number 3 or less.

Developed a reduction construction that preserves cat number.

Characterized infinite graphs allowing indefinite cat evasion.

Abstract

The game of Cat Herding is one in which cat and herder players alternate turns, with the evasive cat moving along non-trivial paths between vertices, and the herder deleting single edges from the graph. Eventually the cat cannot move, and the number of edges deleted is the cat number of the graph. We analyze both when the cat is captured quickly, and when the cat evades capture forever, or for an arbitrarily long time. We develop a reduction construction that retains the cat number of the graph, and classify all (reduced) graphs that have cat number 3 or less as a finite set of graphs. We expand on a logical characterization of infinite Cat Herding on trees to describe all infinite graphs on which the cat can evade capture forever. We also provide a brief characterization of the graphs on which the cat can score arbitrarily high. We conclude by motivating a definition of cat herding ordinals for future research.