On the Structure of Dominating Graphs

Abstract

The k-dominating graph D k (G) of a graph G is defined on the vertex set consisting of dominating sets of G with cardinality at most k, two such sets being adjacent if they differ by either adding or deleting a single vertex. A graph is a dominating graph if it is isomorphic to D k (G) for some graph G and some positive integer k. Answering a question of Haas and Seyffarth for graphs without isolates, it is proved that if G is such a graph of order n ≥ 2 and with G ∼ = D k (G), then k = 2 and G = K 1,n−1 for some n ≥ 4. It is also proved that for a given r there exist only a finite number of r-regular, connected dominating graphs of connected graphs. In particular, C 6 and C 8 are the only dominating graphs in the class of cycles. Some results on the order of dominating graphs are also obtained.

Reconfiguration of Colourings and Dominating Sets in Graphs

“…The concept of k-dominating graphs was introduced by Haas and Seyffarth [34] in 2014. This paper stimulated the work of Alikhani, Fatehi, and Klavžar [1], Mynhardt, Roux, and Teshima [51], Suzuki, Mouawad, and Nishimura [56], and their own follow-up paper [35].…”

“…Haas and Seyffarth [34] considered the question of which graphs are realisable as k-dominating graphs and observed that for n ≥ 4, D 2 (K 1,n−1 ) = K 1,n−1 . Alikhani et al [1] proved that these stars are the only graphs with this property, i.e. if G is a graph of order n with no isolated vertices such that n ≥ 2, δ ≥ 1, and G ∼ = D k (G), then k = 2 and G ∼ = K 1,n−1 for some n ≥ 4.…”

“…* 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.…”

Introduction to Reconfiguration

“…Results have also been obtained for SHORTEST PATH RECONFIGURATION [107], INDEPENDENT SET RECONFIGURATION [108], and TOKEN SLIDING [109], as defined in Section 9.2. Examples of properties that have been studied include chromatic number [109], Hamiltonicity [20,106,[109][110][111][112], and girth [107,113], often in service of determining limits on the classes of graphs represented by reconfiguration graphs [108,[113][114][115]. For a source problem in which the instance is a graph, one can also identify when the reconfiguration graph for an instance is isomorphic to the instance itself [114,115].…”

Introduction to Reconfiguration

“…Results have also been obtained for SHORTEST PATH RECONFIGURATION [107], INDEPENDENT SET RECONFIGURATION [108], and TOKEN SLIDING [109], as defined in Section 9.2. Examples of properties that have been studied include chromatic number [109], Hamiltonicity [20,106,[109][110][111][112], and girth [107,113], often in service of determining limits on the classes of graphs represented by reconfiguration graphs [108,[113][114][115]. For a source problem in which the instance is a graph, one can also identify when the reconfiguration graph for an instance is isomorphic to the instance itself [114,115].…”