Lights Out on graphs
| Autor: | Abraham Berman, Norbert Hungerbühler, Franziska Borer |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Separating invariant
Discrete mathematics Ring (mathematics) General Mathematics Lights Out game Field (mathematics) State (functional analysis) System of linear equations NP‑hard problems Fredholm alternative Lights out Linear algebra FOS: Mathematics Mathematics - Combinatorics Combinatorics (math.CO) Invariant (mathematics) Mathematics |
| Zdroj: | Mathematische Semesterberichte, 68 (2) |
| ISSN: | 1432-1815 0720-728X |
| DOI: | 10.1007/s00591-021-00297-5 |
| Popis: | We model the Lights Out game on general simple graphs in the framework of linear algebra over the field F2. Based upon a version of the Fredholm alternative, we introduce a separating invariant of the game, i.e., an initial state can be transformed into a final state if and only if the values of the invariant of both states agree. We also investigate certain states with particularly interesting properties. Apart from the classical version of the game, we propose several variants, in particular a version with more than only two states (light on, light off), where the analysis relies on systems of linear equations over the ring Zn. Although it is easy to find a concrete solution of the Lights Out problem, we show that it is NP-hard to find a minimal solution. We also propose electric circuit diagrams to actually realize the Lights Out game. Mathematische Semesterberichte, 68 (2) |
| Databáze: | OpenAIRE |
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