Domination and Cut Problems on Chordal Graphs with Bounded Leafage

TL;DR

This paper improves algorithms for domination problems on chordal graphs with bounded leafage, providing simpler and faster solutions, and explores the complexity of cut problems, revealing some are polynomial-time solvable while others are W[1]-hard.

Contribution

It introduces a simpler, faster algorithm for Dominating Set on chordal graphs with bounded leafage and extends it to related problems, also analyzing the complexity of cut problems in this context.

Findings

New $2^{O(l)} n^{O(1)}$ algorithm for Dominating Set.

Polynomial-time algorithm for Multiway Cut with Undeletable Terminals.

W[1]-hardness of MultiCut with Undeletable Terminals.

Abstract

The leafage of a chordal graph G is the minimum integer l such that G can be realized as an intersection graph of subtrees of a tree with l leaves. We consider structural parameterization by the leafage of classical domination and cut problems on chordal graphs. Fomin, Golovach, and Raymond [ESA 2018, Algorithmica 2020] proved, among other things, that Dominating Set on chordal graphs admits an algorithm running in time $2^{O(l^2)} n^{O(1)}$. We present a conceptually much simpler algorithm that runs in time $2^{O(l)} n^{O(1)}$. We extend our approach to obtain similar results for Connected Dominating Set and Steiner Tree. We then consider the two classical cut problems MultiCut with Undeletable Terminals and Multiway Cut with Undeletable Terminals. We prove that the former is W[1]-hard when parameterized by the leafage and complement this result by presenting a simple $n^{O(l)}$-time algorithm. To our surprise, we find that Multiway Cut with Undeletable Terminals on chordal graphs can be solved, in contrast, in $n^{O(1)}$-time.