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Extremal Cat Herding

Rylo Ashmore, Danny Dyer, Rebecca Milley

TL;DR

This work analyzes the extremal behavior of the Cat Herding game on graphs, focusing on graphs with very small cat numbers and on infinite or arbitrarily large cat numbers. It develops pruning techniques that preserve the cat-number to reduce graphs to finite canonical forms, enabling a complete classification of graphs with cat-number at most 2 and a thorough characterization for cat-number 3, including trees and one- or two-cycle structures; it also extends the analysis to infinite graphs by linking cat-win to infinite binary-tree minors and 2-edge-connected subgraphs, and introduces cat-pseudo-win, omega-evadible, and an ordinal framework for transfinite cat numbers to capture unbounded survival scenarios. The approach combines subgraph monotonicity, structural decompositions, and linear-time recognition arguments to obtain sharp classifications and to set the stage for transfinite generalizations. The results provide a rigorous foundation for understanding how graph structure governs capture-length and survival in Cat Herding, with potential algorithmic implications and a path toward ordinal graph parameters. Key contributions include complete classifications for cat-number 1–3, a structural infinite-graph dichotomy via binary-tree minors, and a proposed ordinal extension to quantify unbounded play.

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.

Extremal Cat Herding

TL;DR

This work analyzes the extremal behavior of the Cat Herding game on graphs, focusing on graphs with very small cat numbers and on infinite or arbitrarily large cat numbers. It develops pruning techniques that preserve the cat-number to reduce graphs to finite canonical forms, enabling a complete classification of graphs with cat-number at most 2 and a thorough characterization for cat-number 3, including trees and one- or two-cycle structures; it also extends the analysis to infinite graphs by linking cat-win to infinite binary-tree minors and 2-edge-connected subgraphs, and introduces cat-pseudo-win, omega-evadible, and an ordinal framework for transfinite cat numbers to capture unbounded survival scenarios. The approach combines subgraph monotonicity, structural decompositions, and linear-time recognition arguments to obtain sharp classifications and to set the stage for transfinite generalizations. The results provide a rigorous foundation for understanding how graph structure governs capture-length and survival in Cat Herding, with potential algorithmic implications and a path toward ordinal graph parameters. Key contributions include complete classifications for cat-number 1–3, a structural infinite-graph dichotomy via binary-tree minors, and a proposed ordinal extension to quantify unbounded play.

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.
Paper Structure (6 sections, 32 theorems, 2 equations, 3 figures)

This paper contains 6 sections, 32 theorems, 2 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Graphs of Cat Number at most 2
  3. Graphs of Cat Number 3
  4. Infinite cat herding
  5. Cat-pseudo-win graphs
  6. Conclusion

Key Result

Theorem 1.1

ashmore2024cutscatscompletegraphs If $H$ is a subgraph of $G$, then $\mathrm{cat}(H)\leq \mathrm{cat}(G)$. Further, if $v\in V(H)$, then $\mathrm{cat}(H,v)\leq \mathrm{cat}(G,v)$.

Figures (3)

  • Figure 2.1: Types of (pruned) graphs with cat number 2.
  • Figure 3.1: Types of (pruned) graphs with one cycle and cat number 3. Any dotted edges are optional, provided the resulting edges induce a connected graph. They generally fall into five categories, with (a)-(c) being Triangle-type (a)-(c), (d) being Square-type, and (e) being Pentagon-type.
  • Figure 3.2: Drawing of $T,T_k$ as in \ref{['lem:treeAddPath2']} with edges of $T$ solid and edges of $T_k$ dotted

Theorems & Definitions (63)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 53 more