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Graphs with no long claws: An improved bound for the analog of the Gyárfás' path argument

Romain Bourneuf, Jana Masaříková, Wojciech Nadara, Marcin Pilipczuk

TL;DR

This work strengthens a Gyárfás-path-based structural approach for $S_{t,t,t}$-free graphs by replacing the previously required $O(\log n)$-sized neighborhood deletion with a constant-size set $\mathcal{S}$ of at most $3t+11$. By leveraging a refined selection on the Gyárfás path and the three-in-a-tree theorem, the authors obtain a rigid extended strip decomposition of $G - N[\mathcal{S}]$ in which every particle has weight at most $\frak w(V)/2$, enabling (in particular) streamlined quasi-polynomial-time MWIS results in this graph class. While the full polynomial-time algorithmic payoff does not follow directly, the construction yields immediate simplifications for related MWIS algorithms in $S_{t,t,t}$-free graphs and tightens the structural toolkit available for this family. The result also complements an independent near-identical finding by other researchers, reinforcing the constant-bound refinement of the Gyárfás-path argument in this setting.

Abstract

For a fixed integer $t \geq 1$, a ($t$-)long claw, denoted $S_{t,t,t}$, is the unique tree with three leaves, each at distance exactly $t$ from the vertex of degree three. Majewski et al. [ICALP 2022, ACM ToCT 2024] proved an analog of the Gyárfás' path argument for $S_{t,t,t}$-free graphs: given an $n$-vertex $S_{t,t,t}$-free graph, one can delete neighborhoods of $\mathcal{O}(\log n)$ vertices so that the remainder admits an extended strip decomposition (an appropriate generalization of partition into connected components) into particles of multiplicatively smaller size. This statement has proven to be very useful in designing quasi-polynomial time algorithms for Maximum Weight Independent Set and related problems in $S_{t,t,t}$-free graphs. In this work, we refine the argument of Majewski et al. and show that a constant number of neighborhoods suffice.

Graphs with no long claws: An improved bound for the analog of the Gyárfás' path argument

TL;DR

This work strengthens a Gyárfás-path-based structural approach for -free graphs by replacing the previously required -sized neighborhood deletion with a constant-size set of at most . By leveraging a refined selection on the Gyárfás path and the three-in-a-tree theorem, the authors obtain a rigid extended strip decomposition of in which every particle has weight at most , enabling (in particular) streamlined quasi-polynomial-time MWIS results in this graph class. While the full polynomial-time algorithmic payoff does not follow directly, the construction yields immediate simplifications for related MWIS algorithms in -free graphs and tightens the structural toolkit available for this family. The result also complements an independent near-identical finding by other researchers, reinforcing the constant-bound refinement of the Gyárfás-path argument in this setting.

Abstract

For a fixed integer , a (-)long claw, denoted , is the unique tree with three leaves, each at distance exactly from the vertex of degree three. Majewski et al. [ICALP 2022, ACM ToCT 2024] proved an analog of the Gyárfás' path argument for -free graphs: given an -vertex -free graph, one can delete neighborhoods of vertices so that the remainder admits an extended strip decomposition (an appropriate generalization of partition into connected components) into particles of multiplicatively smaller size. This statement has proven to be very useful in designing quasi-polynomial time algorithms for Maximum Weight Independent Set and related problems in -free graphs. In this work, we refine the argument of Majewski et al. and show that a constant number of neighborhoods suffice.
Paper Structure (5 sections, 10 theorems, 1 equation, 1 figure)

This paper contains 5 sections, 10 theorems, 1 equation, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The result
  4. Overview of the Proof
  5. Proof of \ref{['thm:main-thm']}

Key Result

Theorem 1

Given an $n$-vertex graph $G$ with nonnegative vertex weights, one can in polynomial time find an induced path $Q$ in $G$ such that every connected component of $G-N[V(Q)]$ has weight at most half of the total weight of $V(G)$.

Figures (1)

  • Figure 1: Notations from the proof of \ref{['thm:main-thm']}. $|Q_x|=|Q_y|=|Q_z|=t+1$.

Theorems & Definitions (22)

  • Theorem 1
  • Theorem 2: Sttt_logn23
  • Theorem 3
  • Definition 4: Extended strip decomposition, DBLP:journals/combinatorica/ChudnovskyS10
  • Theorem 5: Chudnovsky, Seymour DBLP:journals/combinatorica/ChudnovskyS10
  • Definition 6
  • Theorem 6
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 12 more