Reconfiguration of Colourings and Dominating Sets in Graphs: a Survey
C.M. Mynhardt, S. Nasserasr

TL;DR
This survey reviews recent results on reconfiguration graphs related to colourings and dominating sets in graphs, discussing their structures, properties, and open problems in the field.
Contribution
It provides a comprehensive overview of various reconfiguration graphs for colourings and dominating sets, highlighting new conjectures and open problems.
Findings
Analysis of the structure of colourings reconfiguration graphs
Discussion of domination reconfiguration graphs and their properties
Identification of open problems and conjectures in the field
Abstract
We survey results concerning reconfigurations of colourings and dominating sets in graphs. The vertices of the -colouring graph of a graph correspond to the proper -colourings of a graph , with two -colourings being adjacent whenever they differ in the colour of exactly one vertex. Similarly, the vertices of the -edge-colouring graph of are the proper -edge-colourings of , where two -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 -dominating graph are the (not necessarily minimal) dominating sets of of cardinality or less, two dominating sets being adjacent in if one can be obtained from the other by…
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Graph Labeling and Dimension Problems
