The orthogonal colouring game

Andres, Stephan Dominique; Huggan, Melissa A.; Inerney, Fionn Mc; Nowakowski, Richard J.

doi:10.1016/j.tcs.2019.07.014

Cited by 9 publications

(14 citation statements)

References 20 publications

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“…of vertices in 2-orbits of V under the action of σ. Andres et al [2] defined an involution σ of G to be strictly matched if (SI 1) the set F 1 (G) induces a (possibly empty) complete graph and, (SI 2) for every v ∈ F 2 (G), we have the (matching) edge vσ(v) ∈ E.…”

Section: Notation and Definitionsmentioning

confidence: 99%

“…Andres et al [2] proved that a graph G admits a strictly matched involution if and only if its vertex set V (G) can be partitioned into a clique C and a set inducing a graph that has a perfect matching M such that:…”

Section: Recognising Graphs Admitting a Strictly Matched Involutionmentioning

confidence: 99%

“…In the scoring variant of the orthogonal colouring game, it is worth noting that playing in the adversary's copy of G may be advantageous in some cases. It may even lead to a win, as the example of the game on the 4-cycle C 4 played with two colours shows, which is won by Bob [2]. The main results of [2] are the introduction and characterisation of a class of graphs, called graphs admitting a strictly matched involution, where it is proven that Bob has a strategy to guarantee a draw in the scoring variant.…”

Section: Introductionmentioning

confidence: 99%

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The Complexity of Two Colouring Games

et al. 2022

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We consider two variants of orthogonal colouring games on graphs. In these games, two players alternate colouring uncoloured vertices (from a choice of $$m\in {\mathbb {N}}$$ m ∈ N colours) of a pair of isomorphic graphs while respecting the properness and the orthogonality of the partial colourings. In the normal play variant, the first player unable to move loses. In the scoring variant, each player aims to maximise their score, which is the number of coloured vertices in their copy of the graph. We prove that, given an instance with partial colourings, both the normal play and the scoring variant of the game are PSPACE-complete. An involution $$\sigma $$ σ of a graph G is strictly matched if its fixed point set induces a clique and $$v\sigma (v)\in E(G)$$ v σ ( v ) ∈ E ( G ) for any non-fixed point $$v\in V(G)$$ v ∈ V ( G ) . Andres et al. (Theor Comput Sci 795:312–325, 2019) gave a solution of the normal play variant played on graphs that admit a strictly matched involution. We prove that recognising graphs that admit a strictly matched involution is NP-complete.

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Section: Notation and Definitionsmentioning

confidence: 99%

Section: Recognising Graphs Admitting a Strictly Matched Involutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Complexity of Two Colouring Games

et al. 2022

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“…Players may gain points in a myriad of ways, all depending on the rules of the game. For example, in the orthogonal colouring game on graphs [1], a player's score is equal to the number of coloured vertices in their copy of the graph at the end of the game, i.e., each player gets one point for each coloured vertex in their copy of the graph. Recently, the papers [12,13,14] have started to build a general theory around scoring games.…”

Section: Introductionmentioning

confidence: 99%

The Largest Connected Subgraph Game

Bensmail

Fioravantes

Inerney

et al. 2021

Graph-Theoretic Concepts in Computer Science

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This paper introduces the largest connected subgraph game played on an undirected graph G. In each round, Alice first colours an uncoloured vertex of G red, and then, Bob colours an uncoloured vertex of G blue, with all vertices initially uncoloured. Once all the vertices are coloured, Alice (Bob, resp.) wins if there is a red (blue, resp.) connected subgraph whose order is greater than the order of any blue (red, resp.) connected subgraph. We first prove that, if Alice plays optimally, then Bob can never win, and define a large class of graphs (called reflection graphs) in which the game is a draw. We then show that determining the outcome of the game is PSPACE-complete, even in bipartite graphs of small diameter, and that recognising reflection graphs is GI-hard. We also prove that the game is a draw in paths if and only if the path is of even order or has at least 11 vertices, and that Alice wins in cycles if and only if the cycle is of odd length. Lastly, we give an algorithm to determine the outcome of the game in cographs in linear time.

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“…A perfect 2-orthogonal colouring is simply called a perfect orthogonal colouring. Perfect orthogonal colourings are of particular importance because they have applications to independent coverings [7] and scoring games [1]. It is now shown that perfect k-orthogonal colourings can build on the applications of (n, k, 1)-transversal designs.…”

Section: Introductionmentioning

confidence: 99%

Orthogonal Colourings of Tensor Graphs

MacKeigan¹

2020

Preprint

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In this paper, perfect k-orthogonal colourings of tensor graphs are studied. First, the problem of determining if a given graph has a perfect 2-orthogonal colouring is reformulated as a tensor subgraph problem. Then, it is shown that if two graphs have a perfect k-orthogonal colouring, then so does their tensor graph. This provides an upper bound on the k-orthogonal chromatic number for general tensor graphs. Lastly, two other conditions for a tensor graph to have a perfect k-orthogonal colouring are given.

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The orthogonal colouring game

Cited by 9 publications

References 20 publications

The Complexity of Two Colouring Games

The Complexity of Two Colouring Games

The Largest Connected Subgraph Game

Orthogonal Colourings of Tensor Graphs

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