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2016 Proving the pressing game conjecture on linear graphs
Eliot Bixby, Toby Flint, István Miklós
Involve 9(1): 41-56 (2016). DOI: 10.2140/involve.2016.9.41

Abstract

The pressing game on black-and-white graphs is the following: given a graph G(V,E) with its vertices colored with black and white, any black vertex v can be pressed, which has the following effect: (1) all neighbors of v change color; i.e., white neighbors become black and vice versa; (2) all pairs of neighbors of v change adjacency; i.e., adjacent pairs become nonadjacent and nonadjacent ones become adjacent; and (3) v becomes a separated white vertex. The aim of the game is to transform G into an all-white, empty graph. It is a known result that the all-white empty graph is reachable in the pressing game if each component of G contains at least one black vertex, and for a fixed graph, any successful transformation has the same number of pressed vertices.

The pressing game conjecture states that any successful pressing sequence can be transformed into any other successful pressing sequence with small alterations. Here we prove the conjecture for linear graphs, also known as paths. The connection to genome rearrangement and sorting signed permutations with reversals is also discussed.

Citation

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Eliot Bixby. Toby Flint. István Miklós. "Proving the pressing game conjecture on linear graphs." Involve 9 (1) 41 - 56, 2016. https://doi.org/10.2140/involve.2016.9.41

Information

Received: 8 April 2013; Revised: 19 January 2015; Accepted: 21 January 2015; Published: 2016
First available in Project Euclid: 22 November 2017

zbMATH: 1328.05120
MathSciNet: MR3438444
Digital Object Identifier: 10.2140/involve.2016.9.41

Subjects:
Primary: 05A05
Secondary: 05CXX

Keywords: Bioinformatics , irreducible Markov chain , pressing game , sorting by reversals

Rights: Copyright © 2016 Mathematical Sciences Publishers

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