The k-Dominating Graph

Abstract

Given a graph G, the k-dominating graph of G, D k (G), is defined to be the graph whose vertices correspond to the dominating sets of G that have cardinality at most k. Two vertices in D k (G) are adjacent if and only if the corresponding dominating sets of G differ by either adding or deleting a single vertex. The graph D k (G) aids in studying the reconfiguration problem for dominating sets. In particular, one dominating set can be reconfigured to another by a sequence of single vertex additions and deletions, such that the intermediate set of vertices at each step is a dominating set if and only if they are in the same connected component of D k (G). In this paper we give conditions that ensure D k (G) is connected.arXiv:1209.5138v2 [math.CO]

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].…”

“…Haas and Seyffarth considered a reconfiguration graph in which solutions are not limited to minimum cardinality dominating sets; their k-dominating graph is a reconfiguration graph under TAR in which each solution is a dominating set of size at most k [114]. For Γ(G), the upper domination number (the maximum cardinality of a minimal dominating set of G), the connectivity of the k-dominating graph has been shown for the following cases:…”

THE k-INDEPENDENT GRAPH OF A GRAPH

“…Regarding to this dominating set reconfiguration problem, recently the k-dominating graph of a graph G has defined in [10]. The k-dominating graph of G, D k (G), is defined to be the graph whose vertices correspond to the dominating sets of G that have cardinality at most k. Two vertices in D k (G) are adjacent if and only if the corresponding dominating sets of G differ by either adding or deleting a single vertex.…”

“…The k-dominating graph of G, D k (G), is defined to be the graph whose vertices correspond to the dominating sets of G that have cardinality at most k. Two vertices in D k (G) are adjacent if and only if the corresponding dominating sets of G differ by either adding or deleting a single vertex. Authors in [10], gave conditions that ensure D k (G) is connected. Also authors in [1] studied this graph, for certain graphs.…”

On the Parameterized Complexity of Reconfiguration Problems

“…Problems for which reconfiguration has been studied include Vertex Colouring [1][2][3][4][5], List Edge-Colouring [6], Independent Set [7,8], Set Cover, Matching, Matroid Bases [8], Satisfiability [9], Shortest Path [10,11], and Dominating Set [12,13]. Most work has been limited to the problem of determining the existence of a reconfiguration sequence between two given solutions; for most NP-complete problems, this problem has been shown to be PSPACE-complete.…”