The Leafed Induced Subtree in chordal and bounded treewidth graphs

TL;DR

This paper develops fixed-parameter tractable algorithms for the Fully Leafed Induced Subtrees problem in graphs with bounded treewidth and chordal graphs, expanding known polynomial solutions.

Contribution

It introduces an FPT algorithm parameterized by treewidth and a polynomial algorithm for chordal graphs for the Fully Leafed Induced Subtrees problem.

Findings

FPT algorithm for graphs with bounded treewidth

Polynomial algorithm for chordal graphs

Generalization of previous NP-complete results

Abstract

In the Fully Leafed Induced Subtrees, one is given a graph $G$ and two integers $a$ and $b$ and the question is to find an induced subtree of $G$ with $a$ vertices and at least $b$ leaves. This problem is known to be NP-complete even when the input graph is $4$-regular. Polynomial algorithms are known when the input graph is restricted to be a tree or series-parallel. In this paper we generalize these results by providing an FPT algorithm parameterized by treewidth. We also provide a polynomial algorithm when the input graph is restricted to be a chordal graph.