The lights out game on directed graphs

Dettling, T. Elise; Parker, Darren B.

doi:10.2140/involve.2026.19.33

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The lights out game on directed graphs

T. Elise Dettling and Darren B. Parker

Vol. 19 (2026), No. 1, 33–50

DOI: 10.2140/involve.2026.19.33

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 $ℤ_{k}$ for $k \geq 2$ . 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 $ℤ_{k}$ . 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 certain tournaments with feedback arc sets that arc-induce directed paths or directed star digraphs.

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Keywords

lights out, light-switching game, feedback arc sets, linear algebra

Mathematical Subject Classification

Primary: 05C20, 05C50, 05C57, 05C78

Secondary: 05C40

Milestones

Received: 9 June 2023

Revised: 8 August 2024

Accepted: 14 August 2024

Published: 25 January 2026

Communicated by Kenneth S. Berenhaut

Authors

T. Elise Dettling
Department of Mathematics Grand Valley State University Allendale, MI United States

Darren B. Parker
Department of Mathematics Grand Valley State University Allendale, MI United States

Vol. 19, No. 1, 2026