[Paper Review] The complexity of dominating set reconfiguration

Arash Haddadan, Takehiro Ito+5 more|arXiv (Cornell University)|Mar 3, 2015

Advanced Graph Theory Research8 references4 citations

TL;DR

This paper investigates the complexity of reconfiguring dominating sets in graphs under a cardinality constraint, proving the problem is PSPACE-complete for planar, bounded bandwidth, split, and bipartite graphs. It introduces a general linear-time algorithmic framework applicable to cographs, trees, and interval graphs, showing that reconfiguration sequences can be constructed in O(n) steps, establishing a sharp boundary between tractable and intractable graph classes for this problem.

ABSTRACT

Suppose that we are given two dominating sets $D_s$ and $D_t$ of a graph $G$ whose cardinalities are at most a given threshold $k$. Then, we are asked whether there exists a sequence of dominating sets of $G$ between $D_s$ and $D_t$ such that each dominating set in the sequence is of cardinality at most $k$ and can be obtained from the previous one by either adding or deleting exactly one vertex. This problem is known to be PSPACE-complete in general. In this paper, we study the complexity of this decision problem from the viewpoint of graph classes. We first prove that the problem remains PSPACE-complete even for planar graphs, bounded bandwidth graphs, split graphs, and bipartite graphs. We then give a general scheme to construct linear-time algorithms and show that the problem can be solved in linear time for cographs, trees, and interval graphs. Furthermore, for these tractable cases, we can obtain a desired sequence such that the number of additions and deletions is bounded by $O(n)$, where $n$ is the number of vertices in the input graph.

Motivation & Objective

• To determine the computational complexity of reconfiguring dominating sets under a cardinality threshold k across various graph classes.
• To identify structural properties that distinguish PSPACE-complete instances from polynomial-time solvable ones.
• To develop a general algorithmic framework for efficiently computing reconfiguration sequences in tractable graph classes.
• To establish tight bounds on the length of reconfiguration sequences in these tractable cases.

Proposed method

• Proved PSPACE-completeness via reductions from NP-hard dominating set instances, preserving reconfigurability.
• Proposed a general algorithmic scheme based on identifying a canonical dominating set in cographs, trees, and interval graphs.
• For interval graphs, used interval representations and vertex labeling based on right endpoints to construct a canonical dominating set.
• For trees, applied a greedy labeling strategy from left to right using rightmost vertex selection in closed neighborhoods.
• For cographs, leveraged their modular decomposition and P4-free structure to define a canonical dominating set.
• Demonstrated that reconfiguration sequences from any valid dominating set to the canonical one require O(n) operations, ensuring linear-time solvability.

Experimental results

Research questions

• RQ1 What is the complexity of dominating set reconfiguration on planar graphs, bounded bandwidth graphs, split graphs, and bipartite graphs?
• RQ2 Can a general algorithmic framework be designed for reconfiguration in graph classes admitting a canonical dominating set?
• RQ3 Is the length of the reconfiguration sequence bounded by O(n) for tractable graph classes like cographs, trees, and interval graphs?
• RQ4 What structural properties of a graph determine whether the reconfiguration problem is PSPACE-complete or solvable in linear time?

Key findings

• The dominating set reconfiguration problem is PSPACE-complete even for planar graphs, bounded bandwidth graphs, split graphs, and bipartite graphs.
• A general linear-time algorithmic framework was developed for cographs, trees, and interval graphs, enabling efficient reconfiguration sequence computation.
• For cographs, trees, and interval graphs, any valid reconfiguration sequence between two dominating sets of size at most k can be constructed in O(n) steps.
• The canonical dominating set in interval graphs is constructed via interval endpoint-based labeling and proven to be minimum and universal via reconfiguration.
• The problem remains PSPACE-complete for chordal graphs, highlighting a sharp complexity boundary between interval graphs and their super-class.
• The reconfiguration sequence length is bounded by O(n) for all tractable cases, implying optimal or near-optimal efficiency in terms of vertex operations.