Reconfiguring minimum  dominating sets: the γ-graph of a tree

Edwards, Michelle; MacGillivray, Gary; Nasserasr, Shahla

doi:10.7151/dmgt.2044

Cited by 6 publications

(13 citation statements)

References 3 publications

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“…Edwards, MacGillivray, and Nasserasr [2] determined the following upper bounds on the order, diameter, and maximum degree of both jump and slide γ-graphs of trees, which answers open questions discussed by Fricke et al [4]. A support vertex is the unique neighbour of a vertex of degree one.…”

Section: Introductionmentioning

confidence: 66%

“…We see that the maximum degree and diameter of γ-graphs of trees are linear in the number of vertices. Edwards et al [2] demonstrated that the bounds in (i) are sharp for an infinite family of trees, stated that no known tree has a γ-graph whose diamter exceeds half the bound in (ii), and showed that |V (S(T, γ)| > 2 γ(T ) for infinitely many trees. Lemańska and Żyliński [5] determined the following tight bounds on the diameter.…”

Section: Theorem 2 [2]mentioning

confidence: 99%

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A Bipartite Graph That Is Not the $γ$-Graph of a Bipartite Graph

van Bommel

2020

Preprint

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For a graph G = (V, E), the γ-graph of G is the graph whose vertex set is the collection of minimum dominating sets, or γ-sets of G, and two γ-sets are adjacent if they differ by a single vertex and the two different vertices are adjacent in G. An open question in γ-graphs is whether every bipartite graph is the γ-graph of some bipartite graph. We answer this question in the negative by demonstrating that K2,3 is not the γ-graph of any bipartite graph.

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Section: Introductionmentioning

confidence: 66%

Section: Theorem 2 [2]mentioning

confidence: 99%

A Bipartite Graph That Is Not the $γ$-Graph of a Bipartite Graph

van Bommel

2020

Preprint

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“…Edwards, MacGillivray, and Nasserasr [28] obtained results which hold for jump and slide γ-graphs; we report their results in Theorem 4.6.…”

Section: Jump γ-Graphsmentioning

confidence: 78%

“…Edwards et al [28] investigated the order, diameter, and maximum degree of jump and slide γ-graphs of trees, providing answers to questions posed in [31].…”

Section: Slide γ-Graphsmentioning

confidence: 99%

Reconfiguration of Colourings and Dominating Sets in Graphs

Nasserasr¹

2019

50 Years of Combinatorics, Graph Theory, and Computing

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We survey results concerning reconfigurations of colourings and dominating sets in graphs. The vertices of the k-colouring graph C k (G) of a graph G correspond to the proper k-colourings of a graph G, with two k-colourings being adjacent whenever they differ in the colour of exactly one vertex. Similarly, the vertices of the k-edge-colouring graph EC k (G) of g are the proper k-edge-colourings of G, where two k-edge-colourings are adjacent if one can be obtained from the other by switching two colours along an edge-Kempe chain, i.e., a maximal two-coloured alternating path or cycle of edges.The vertices of the k-dominating graph D k (G) are the (not necessarily minimal) dominating sets of G of cardinality k or less, two dominating sets being adjacent in D k (G) if one can be obtained from the other by adding or deleting one vertex. On the other hand, when we restrict the dominating sets to be minimum dominating sets, for example, we obtain different types of domination reconfiguration graphs, depending on whether vertices are exchanged along edges or not.We consider these and related types of colouring and domination reconfiguration graphs. Conjectures, questions and open problems are stated within the relevant sections. * Supported by the Natural Sciences and Engineering Research Council of Canada.1 We use the term connectedness instead of connectivity when referring to the question of whether a graph is connected or not, as the latter term refers to a specific graph parameter.

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“…We present a general algorithm to determine the γ-graph of a tree. We first state the following results of Edwards, MacGillivray, and Nasserasr [3] which will aid in verifying the algorithm. If D is a γ-set of a rooted tree (T, c), then the height of D, denoted ht T (D), is The algorithm DOMSET, developed by Cockayne, Goodman, and Hedetniemi [6] finds a minimum dominating set of a tree in linear time.…”

Section: Computing γ-Graphs Of Treesmentioning

confidence: 99%

γ-Graphs of Trees

Finbow

Bommel

2019

Algorithms

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For a graph G = (V, E), the γ-graph of G, denoted G(γ) = (V (γ), E(γ)), is the graph whose vertex set is the collection of minimum dominating sets, or γ-sets of G, and two γ-sets are adjacent in G(γ) if they differ by a single vertex and the two different vertices are adjacent in G. In this paper, we consider γ-graphs of trees. We develop an algorithm for determining the γ-graph of a tree, characterize which trees are γ-graphs of trees, and further comment on the structure of γ-graphs of trees and its connections with Cartesian product graphs, the set of graphs which can be obtained from the Cartesian product of graphs of order at least two.

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Reconfiguring minimum dominating sets: the γ-graph of a tree

Cited by 6 publications

References 3 publications

A Bipartite Graph That Is Not the $γ$-Graph of a Bipartite Graph

A Bipartite Graph That Is Not the $γ$-Graph of a Bipartite Graph

Reconfiguration of Colourings and Dominating Sets in Graphs

γ-Graphs of Trees

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