Undirected edge geography games on grids

TL;DR

This work completely characterizes the rooted Undirected Edge Geography on grid graphs $G_{m\times n}$ by showing that, with $d=\gcd(m+1,n+1)$, the position $(G_{m\times n},v)$ is a P-position precisely when $d \nmid a$ and $d \nmid b$, generalizing the corner result. The main method combines geometric trails (a 90-degree trail for N-positions and a 180-degree trail for P-positions) with region-labeling and the construction of even kernels; explicit $S_k$ families provide concrete kernel realizations. The results yield complete winning strategies: a closed $90$-degree trail leads to a winning move for the next player, while a $180$-degree trail yields a P-position via an even kernel. Overall, the paper advances understanding of geography-type games on grids by delivering exact classifications and constructive strategies for all rooted positions.

Abstract

The undirected edge geography is a two-player combinatorial game on an undirected rooted graph. The players alternatively perform a move consisting of choosing an edge incident to the root vertex, removing the chosen edge, and marking the other endpoint as a new root vertex. The first player who cannot perform a move is the loser. In this paper, we are interested in the undirected edge geography game on the grid graph $P_m\square P_n$. We completely determine whether the root vertex is a winning position (N-position) or a losing position (P-position). Moreover, we give a winning strategy for the winner.

Figures (6)

• Figure 1: The vertex $(2,2)$ of $G_{5,3}$ has exactly two trails that there are two straight line segments touching $(2,2)$.
• Figure 2: An embedding of multiple copies of $G_{3,2}$ on the rectangle $[0,8]\times[0,9]$ where $(h,k)=(1,1)$.
• Figure 3: (a) A proper labeling of the regions obtained from the extended trail at the vertex $(3,4)$ of $G_{11\times 8}$. (b) The result after switching the labels of the regions on the right of the extended segments. The positive regions are shown in blue. All unbounded regions are negative.
• Figure 4: All possibilities of the neighbors on a trail.
• Figure 5: (a) An even kernel (black vertices) of $G_{11\times 8}$ containing $(2,4)$ (blue circle). (b) An even kernel (black vertices) of $G_{11\times 8}-uv$ where $u=(3,4)$ (blue circle) and $v=(2,4)$ (red circle). The positive regions are shown in blue.

Theorems & Definitions (19)

• Theorem 1: Fraenkel1993
• Theorem 2
• Theorem 3
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof
• Theorem 7