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The Leafed Induced Subtree in chordal and bounded treewidth graphs

Julien Baste

TL;DR

This work addresses the Restricted Leafed Induced Subtree problem and its fully leafed variant by delivering two main results. It proves a polynomial-time dynamic programming algorithm for chordal graphs, leveraging a clique-ward nice tree decomposition and a crucial bound that partial solutions interact with bag boundaries on at most two vertices. It also provides a single-exponential FPT algorithm parameterized by treewidth, built on Bodlaender et al.'s weighted-partition framework to manage solution interfaces across a tree decomposition, achieving time $2^{O(\mathbf{tw})} \cdot \mathbf{tw}^{O(1)} \cdot n^7$. Together, these contributions extend tractability of leaf-constrained induced subtree problems to chordal graphs and graphs of bounded treewidth, with implications for related subgraph problems and practical applications.

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.

The Leafed Induced Subtree in chordal and bounded treewidth graphs

TL;DR

This work addresses the Restricted Leafed Induced Subtree problem and its fully leafed variant by delivering two main results. It proves a polynomial-time dynamic programming algorithm for chordal graphs, leveraging a clique-ward nice tree decomposition and a crucial bound that partial solutions interact with bag boundaries on at most two vertices. It also provides a single-exponential FPT algorithm parameterized by treewidth, built on Bodlaender et al.'s weighted-partition framework to manage solution interfaces across a tree decomposition, achieving time . Together, these contributions extend tractability of leaf-constrained induced subtree problems to chordal graphs and graphs of bounded treewidth, with implications for related subgraph problems and practical applications.

Abstract

In the Fully Leafed Induced Subtrees, one is given a graph and two integers and and the question is to find an induced subtree of with vertices and at least leaves. This problem is known to be NP-complete even when the input graph is -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.
Paper Structure (5 sections, 7 theorems, 1 equation)

This paper contains 5 sections, 7 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Chordal graphs
  4. A single exponential algorithm parameterized by treewidth
  5. Conclusion

Key Result

Theorem 1

Restricted Leafed Induced Subtree on an instance $(G,v_0,{a},{b})$ where $G$ is a chordal graph can be solved in time ${\mathcal{O}}({a}^2\cdot n^4)$, where $n$ is the number of vertices of $G$.

Theorems & Definitions (11)

  • Claim 1
  • proof
  • Theorem 1
  • Proposition 1: Bodlaender et al. BoCyKrNe2015
  • Corollary 1
  • proof
  • Proposition 2: Bodlaender et al. BoCyKrNe2015
  • Corollary 2
  • Theorem 2: Bodlaender et al. BoCyKrNe2015
  • Theorem 3
  • ...and 1 more