Reconfiguration of Colourings and Dominating Sets in Graphs

Abstract

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.

Dominating sets reconfiguration under token sliding

“…In particular, they proved that R γ (G) is connected and bipartite if G is a tree. For a more complete overview, the reader is referred to [28].…”

Homomorphism complexes, reconfiguration, and homotopy for directed graphs

“…This has practical applications when exact counting of solutions is not possible in reasonable time, in which case Markov chain simulation can be used. For these questions the connectedness, diameter, realizability, and algorithmic properties of the configuration graph are typically studied (see [58] for a survey). In our context a pair of homomorphisms of directed graphs 𝑓 , 𝑔 : 𝐺 → 𝐻 will be considered adjacent if 𝑓 and 𝑔 agree on all but one vertex, defining a reconfiguration graph that corresponds to the -skeleton of − −− → Hom(𝐺, 𝐻).…”

Hamilton Paths in Dominating Graphs of Trees and Cycles

“…Most of these focus on conditions on k that ensure that D k (G) is connected. Two recent surveys of reconfiguration of dominating sets are [1] and [9].…”

“…Nishimura [8] examines reconfiguration from an algorithmic perspective and considers complexity questions in a wide range of reconfiguration settings. Reconfiguration of graph coloring problems and dominating set problems are surveyed in a recent paper of Mynhardt and Nasserasr [9].…”