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A New Dominating Set Game on Graphs

Sean Fiscus, Glenn Hurlbert, Eric Myzelev, Travis Pence

TL;DR

This paper introduces the snooker domination (SD) game on finite connected graphs, where two players alternately select vertices until the chosen set is dominating and the last mover wins. It develops an involution-based framework, notably 3-involution, to classify large graph families as P-positions and uses graph-constructive tools (graph power, Cartesian product, join, dangling, bridging) to transfer results to rich classes such as cubes, multidimensional grids, caterpillars, the Petersen graph, sunlets, and abelian Cayley graphs. The authors precisely characterize winner behavior on cycles and paths (via PD/CD variants) and extend the analysis to the group setting through Cayley graphs, providing a broad spectrum of exact determinations (P/N) and robust strategies. They conclude with open problems and conjectures, outlining avenues for extending the framework to other graph classes and related variants, thereby offering both structural theory and practical winner-determination methods for graph-based games.

Abstract

We introduce a new two-player game on graphs, in which players alternate choosing vertices until the set of chosen vertices forms a dominating set. The last player to choose a vertex is the winner. The game fits into the scheme of several other known games on graphs. We characterize the paths and cycles for which the first player has the winning strategy. We also create tools for combining graphs in various ways (via graph powers, Cartesian products, graph joins, and other methods) for building a variety of graphs whose games are won by the second player, including cubes, multidimensional grids with an odd number of vertices, most multidimensional toroidal grids, various trees such as specialized caterpillars, the Petersen graph, and others. Finally, we extend the game to groups and show that the second player wins the game on abelian groups of even order with canonical generating set, among others.

A New Dominating Set Game on Graphs

TL;DR

This paper introduces the snooker domination (SD) game on finite connected graphs, where two players alternately select vertices until the chosen set is dominating and the last mover wins. It develops an involution-based framework, notably 3-involution, to classify large graph families as P-positions and uses graph-constructive tools (graph power, Cartesian product, join, dangling, bridging) to transfer results to rich classes such as cubes, multidimensional grids, caterpillars, the Petersen graph, sunlets, and abelian Cayley graphs. The authors precisely characterize winner behavior on cycles and paths (via PD/CD variants) and extend the analysis to the group setting through Cayley graphs, providing a broad spectrum of exact determinations (P/N) and robust strategies. They conclude with open problems and conjectures, outlining avenues for extending the framework to other graph classes and related variants, thereby offering both structural theory and practical winner-determination methods for graph-based games.

Abstract

We introduce a new two-player game on graphs, in which players alternate choosing vertices until the set of chosen vertices forms a dominating set. The last player to choose a vertex is the winner. The game fits into the scheme of several other known games on graphs. We characterize the paths and cycles for which the first player has the winning strategy. We also create tools for combining graphs in various ways (via graph powers, Cartesian products, graph joins, and other methods) for building a variety of graphs whose games are won by the second player, including cubes, multidimensional grids with an odd number of vertices, most multidimensional toroidal grids, various trees such as specialized caterpillars, the Petersen graph, and others. Finally, we extend the game to groups and show that the second player wins the game on abelian groups of even order with canonical generating set, among others.

Paper Structure

This paper contains 16 sections, 17 theorems, 3 figures.

Table of Contents

  1. Introduction
  2. History
  3. Game Theory Basics
  4. SD and PD Game Notations
  5. Involutions
  6. Graph constructions
  7. Our results
  8. Key Lemmas
  9. SD games
  10. PD games
  11. Proofs
  12. Proof of Theorem \ref{['t:InvoP']}
  13. Proof of Theorem \ref{['t:DomGameCycle']}
  14. Proof of Theorem \ref{['t:DomGamePath']}
  15. Other specific graphs
  16. ...and 1 more sections

Key Result

Lemma 1

If ${\cal Q}$ is a 3-involutionary position, then ${\cal Q}\in {\sf P}$. In particular, if the graph $G$ is 3-involutionary then ${\mathbb D}(G)\in{\sf P}$.

Figures (3)

  • Figure 1: The 7-sunlet graph $S_7$.
  • Figure 2: The internally 3-regular caterpillar $(P_3,x)$--$(P_3,y)$.
  • Figure 3: The Petersen graph $P=K(5,2)$ with its vertex labels shown.

Theorems & Definitions (35)

  • Lemma 1: Involution Lemma
  • proof
  • Corollary 2
  • proof
  • Corollary 3
  • proof
  • Example 4
  • Example 5
  • Proposition 6
  • Theorem 7
  • ...and 25 more