Cuts, Cats, and Complete Graphs

TL;DR

This paper introduces the game of Cat Herding, analyzes its properties on various graph classes, and determines optimal strategies and scores, including a closed-form solution for complete graphs.

Contribution

It provides the first detailed analysis of Cat Herding, deriving exact scores and strategies for paths, cycles, stars, wheels, and complete graphs.

Findings

Exact cat number for paths, cycles, stars, wheels

Optimal strategies for complete graphs

Closed-form formula for cat number of complete graphs

Abstract

We introduce the game of Cat Herding, where an omnipresent herder slowly cuts down a graph until an evasive cat player has nowhere to go. The number of cuts made is the score of a game, and we study the score under optimal play. In this paper, we begin by deriving some general results, and then we determine the precise cat number for paths, cycles, stars, and wheels. Finally, we identify an optimal Cat and Herder strategy on complete graphs, while providing both a recurrence and closed form for $cat(K_n)$.