The Avoider-Enforcer game on hypergraphs of rank 3

TL;DR

The paper develops a general framework for Avoider-Enforcer games on hypergraphs, introducing the last-move paradigm and showing that a hypergraph’s outcome is determined by who moves last. It proves that disjoint unions have outcomes determined by their components and provides monotonicity and pairing tools, along with a duality via transversals. It then completely solves the AE-game for all rank-2 hypergraphs (graphs) and initiates the rank-3 linear case by giving a full structural characterization when Avoider is the last player, with polynomial-time recognition. The results yield practical algorithms to decide outcomes and highlight key structures—cycles, prisms, nunchakus, and leaf-edge reductions—that govern strategy. Overall, the work lays a solid foundation for AE-games on small-rank hypergraphs and suggests clear directions for higher-rank generalizations.

Abstract

In the Avoider-Enforcer convention of positional games, two players, Avoider and Enforcer, take turns selecting vertices from a hypergraph H. Enforcer wins if, by the time all vertices of H have been selected, Avoider has completely filled an edge of H with her vertices; otherwise, Avoider wins. In this paper, we first give some general results, in particular regarding the outcome of the game and disjoint unions of hypergraphs. We then determine which player has a winning strategy for all hypergraphs of rank 2, and for linear hypergraphs of rank 3 when Avoider plays the last move. The structural characterisations we obtain yield polynomial-time algorithms.

Figures (8)

• Figure 1: Examples of hypergraphs for each possible outcome.
• Figure 2: Counterexamples to the converse of the first (left) and the second (right) assertions of Proposition \ref{['prop:lastmove']}. As in all figures from now, we represent edges of size 2 with a straight line and edges of size 3 with a "claw" shape.
• Figure 3: Some elementary graphs.
• Figure 5: Some elementary linear hypergraphs of rank 3.
• Figure 6: Top: if $x$ is of degree 2 in $N$ (left) then $N^{+x}$ contains two disjoint nunchakus (right). Bottom: if $x$ is of degree 1 in $N$ (left) then $N^{+x}$ contains two nunchakus sharing a 2-edge (right).

Theorems & Definitions (61)

• Theorem 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Lemma 2.5
• proof
• Theorem 2.6
• proof