Reconfiguration of dominating sets

Abstract

Abstract. We explore a reconfiguration version of the dominating set problem, where a dominating set in a graph G is a set S of vertices such that each vertex is either in S or has a neighbour in S. In a reconfiguration problem, the goal is to determine whether there exists a sequence of feasible solutions connecting given feasible solutions s and t such that each pair of consecutive solutions is adjacent according to a specified adjacency relation. Two dominating sets are adjacent if one can be formed from the other by the addition or deletion of a single vertex. For various values of k, we consider properties of D k (G), the graph consisting of a vertex for each dominating set of size at most k and edges specified by the adjacency relation. Addressing an open question posed by Haas and Seyffarth, we demonstrate that D Γ (G)+1 (G) is not necessarily connected, for Γ (G) the maximum cardinality of a minimal dominating set in G. The result holds even when graphs are constrained to be planar, of bounded tree-width, or b-partite for b ≥ 3. Moreover, we construct an infinite family of graphs such that D γ(G)+1 (G) has exponential diameter, for γ(G) the minimum size of a dominating set. On the positive side, we show that Dn−m(G) is connected and of linear diameter for any graph G on n vertices having at least m + 1 independent edges.

Reconfiguring Independent Sets in Claw-Free Graphs

“…The reconfiguration paradigm has recently been applied to a number of algorithmic problems: vertex coloring [1,2,3,4,5], list-edge coloring [7], clique, set cover, integer programming, matching, spanning tree, matroid bases [8], block puzzles [9], satisfiability [10], independent set [9,8,11], shortest paths [12,13,14], and dominating set [15]; recently also in the setting of parameterized complexity [16]. A recent survey [17] gives a good introduction to this area of research.…”

Introduction to Reconfiguration

“…bipartite and chordal, when k ≥ Γ(G) + 1 [114], • k = n − µ and there is a matching of cardinality at least µ + 1 [127], • k = Γ(G) + 1 for certain classes of well-covered graphs [128], • k = Γ(G) + 1 for graphs that are both perfect and irredundant perfect [128].…”

“…In contrast, the k-dominating graph can be disconnected for k = Γ(G) + 1, even for planar, bounded tree-width, or b-partite for b ≥ 3 [127]. In addition, there is an infinite family of graphs with exponential diameter for γ(G) + 1 [127].…”

Introduction to Reconfiguration

“…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: In contrast, the k-dominating graph can be disconnected for k = Γ(G) + 1, even for planar, bounded tree-width, or b-partite for b ≥ 3 [126]. In addition, there is an infinite family of graphs with exponential diameter for γ(G) + 1 [126]. Under TAR, the reachability problem is PSPACE-complete even for bounded bandwidth, split graphs, planar graphs, and bipartite graphs [50], whereas linear-time algorithms have been developed for cographs, trees, and interval graphs [50].…”