32 research outputs found
Reconfiguration of Dominating Sets
We explore a reconfiguration version of the dominating set problem, where a
dominating set in a graph is a set of vertices such that each vertex is
either in or has a neighbour in . In a reconfiguration problem, the goal
is to determine whether there exists a sequence of feasible solutions
connecting given feasible solutions and 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 , we consider properties of , the graph
consisting of a vertex for each dominating set of size at most and edges
specified by the adjacency relation. Addressing an open question posed by Haas
and Seyffarth, we demonstrate that is not necessarily
connected, for the maximum cardinality of a minimal dominating set
in . The result holds even when graphs are constrained to be planar, of
bounded tree-width, or -partite for . Moreover, we construct an
infinite family of graphs such that has exponential
diameter, for the minimum size of a dominating set. On the positive
side, we show that is connected and of linear diameter for any
graph on vertices having at least independent edges.Comment: 12 pages, 4 figure
TS-Reconfiguration of Dominating Sets in circle and circular-arc graphs
We study the dominating set reconfiguration problem with the token sliding
rule. It consists, given a graph G=(V,E) and two dominating sets D_s and D_t of
G, in determining if there exists a sequence S= of
dominating sets of G such that for any two consecutive dominating sets D_r and
D_{r+1} with r<t, D_{r+1}=(D_r\ u) U v, where uv is an edge of G.
In a recent paper, Bonamy et al studied this problem and raised the following
questions: what is the complexity of this problem on circular arc graphs? On
circle graphs? In this paper, we answer both questions by proving that the
problem is polynomial on circular-arc graphs and PSPACE-complete on circle
graphs.Comment: This work was supported by ANR project GrR (ANR-18-CE40-0032) and
submitted to the conference WADS 202
Shortest Dominating Set Reconfiguration under Token Sliding
In this paper, we present novel algorithms that efficiently compute a
shortest reconfiguration sequence between two given dominating sets in trees
and interval graphs under the Token Sliding model. In this problem, a graph is
provided along with its two dominating sets, which can be imagined as tokens
placed on vertices. The objective is to find a shortest sequence of dominating
sets that transforms one set into the other, with each set in the sequence
resulting from sliding a single token in the previous set. While identifying
any sequence has been well studied, our work presents the first polynomial
algorithms for this optimization variant in the context of dominating sets.Comment: To appear at FCT 2023 (Fundamentals of Computation Theory