Parameterized Domination in Circle Graphs

TL;DR

This paper investigates the parameterized complexity of various domination problems in circle graphs, revealing hardness results and identifying a polynomial-time solvable case, along with an FPT algorithm for tree isomorphism problems.

Contribution

It establishes W[1]-hardness for multiple domination problems, finds a polynomial-time solution for Connected Acyclic Dominating Set, and develops a subexponential FPT algorithm for tree isomorphism in circle graphs.

Findings

W[1]-hardness of several domination problems in circle graphs

Polynomial-time solvability of Connected Acyclic Dominating Set

Subexponential FPT algorithm for tree isomorphism in circle graphs

Abstract

A circle graph is the intersection graph of a set of chords in a circle. Keil [Discrete Applied Mathematics, 42(1):51-63, 1993] proved that Dominating Set, Connected Dominating Set, and Total Dominating Set are NP-complete in circle graphs. To the best of our knowledge, nothing was known about the parameterized complexity of these problems in circle graphs. In this paper we prove the following results, which contribute in this direction: - Dominating Set, Independent Dominating Set, Connected Dominating Set, Total Dominating Set, and Acyclic Dominating Set are W[1]-hard in circle graphs, parameterized by the size of the solution. - Whereas both Connected Dominating Set and Acyclic Dominating Set are W[1]-hard in circle graphs, it turns out that Connected Acyclic Dominating Set is polynomial-time solvable in circle graphs. - If T is a given tree, deciding whether a circle graph has a dominating set isomorphic to T is NP-complete when T is in the input, and FPT when parameterized by |V(T)|. We prove that the FPT algorithm is subexponential.