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arXiv · 2306.06017

The Lights Out Game on Directed Graphs

T. Elise Dettling·Darren B. Parker
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Abstract

We study a version of the lights out game played on directed graphs. For a digraph $D$, we begin with a labeling of $V(D)$ with elements of $\mathbb{Z}_k$ for $k \ge 2$. When a vertex $v$ is toggled, the labels of $v$ and any vertex that $v$ dominates are increased by 1 mod $k$. The game is won when each vertex has label 0. We say that $D$ is $k$-Always Winnable (also written $k$-AW) if the game can be won for every initial labeling with elements of $\mathbb{Z}_k$. We prove that all acyclic digraphs are $k$-AW for all $k$, and we reduce the problem of determining whether a graph is $k$-AW to the case of strongly connected digraphs. We then determine winnability for tournaments with a minimum feedback arc set that arc-induces a directed path or directed star digraph.

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BibTeXRIS

T. Elise Dettling, Darren B. Parker. 2023-06-09. The Lights Out Game on Directed Graphs. https://doi.org/10.2140/involve.2026.19.33

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