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Representative Papers

Reuniting χ-boundedness with polynomial χ-boundedness

Oct 17, 2023arXiv.org

This paper investigates the structural relationship between χ-boundedness and polynomial χ-boundedness by introducing and systematically studying “Pollyanna graph classes”—graph classes whose intersection with any χ-bounded class remains polynomially χ-bounded. Method: Employing combinatorial graph theory, induced-subgraph coloring analysis, and asymptotic growth-rate comparison, the authors develop a necessary and sufficient condition framework for Pollyanna property. Contribution/Results: They prove that several fundamental graph classes—including perfect graphs and chordal graphs—are Pollyanna; moreover, they construct the first explicit non-Pollyanna graph class, establishing the nontriviality of the property. These results provide a novel classification tool and decision paradigm for χ-boundedness theory, enriching its structural foundations and offering new criteria for distinguishing polynomial from general χ-bounded behavior.

4 citations2 influentialRead paper

Excluding an induced wheel minor in graphs without large induced stars

Jun 10, 2025

This paper addresses a conjecture by Dallard et al. concerning the boundedness of tree-independence number: for any positive integer $d$ and planar graph $H$, is the tree-independence number bounded for the class of graphs excluding $K_{1,d}$ as an induced subgraph and $H$ as an induced minor? The focus is specifically on $H$ being a $k$-wheel ($k geq 3$). Method: We generalize the notion of bramble to the tree-independence framework and establish a fundamental connection between excluding $k$-wheels as induced minors and structural boundedness. Our approach integrates generalized bramble theory, structural analysis of graph minors, and tree decomposition characterizations. Contribution/Results: We prove that graphs excluding $K_{1,d}$ and $k$-wheels as induced minors have bounded tree-independence number. Consequently, several NP-hard problems—including Maximum Independent Set—become polynomial-time solvable on this graph class. Furthermore, we design a polynomial-time algorithm to detect $k$-wheel induced minors.

1 citations1 influentialRead paper

Variants of Merge-Width and Applications

Feb 27, 2026

Existing theoretical frameworks for unified graph parameters—such as treewidth, clique-width, and twin-width—lack robustness and algorithmic support. This work focuses on merge-width and its variants, establishing deep connections to sparse graph theory, neighborhood covers, and bounded expansion graph classes through characterizations based on vertex orderings and logical definability. We present the first set of equivalent definitions for merge-width, design the first non-trivial approximation algorithm running in $n^{O(1)} \cdot 2^n$ time, and demonstrate its existence in structures with constant-overlap neighborhood covers and quasi-isometric embeddings. These contributions collectively establish the robustness and algorithmic applicability of merge-width as a graph parameter.

1 citationsRead paper

Basis Number of Graphs Excluding Minors

Jan 08, 2026

This study investigates whether the basis number of graphs excluding a fixed minor $H$ is bounded—that is, whether their cycle space admits a generating set of cycles in which each edge appears at most a constant number of times. By integrating graph minor exclusion theory with tree and path decomposition techniques based on the Bojańczyk–Pilipczuk framework, the work establishes the first connection between basis number and minor exclusion. The main contribution is a proof that every $H$-minor-free graph has a bounded basis number, together with an explicit polynomial upper bound of the form $O(|H|^c)$, where $c$ is an absolute constant. This result significantly extends beyond prior bounds that were restricted to graphs embeddable on surfaces.

1 citationsRead paper

On graphs coverable by chubby shortest paths

Mar 04, 2025

This paper investigates the structural properties of graphs covered by “fat shortest paths”—i.e., graphs in which every vertex lies within distance at most ρ of some shortest path. We introduce the ρ-fault-tolerant path cover model and establish quantitative coarse-geometric relationships between it and classical graph parameters: such graphs are coarsely quasi-isometric to graphs of pathwidth $k^{O(k)}$ and maximum degree $O(k)$. Furthermore, we construct a path decomposition with spherical covering property: each bag is covered by $k^{O(k)}$ balls of radius at most $2 ho$. This decomposition paradigm bridges coarse geometry and path decomposition theory for the first time, yielding a foundational framework for designing efficient approximation algorithms for distance-constrained combinatorial optimization problems—including Maximum Distance-Independent Set and Minimum Distance-Dominating Set.

1 citationsRead paper

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Latest Papers

Coloring graphs with no long induced path

Sep 08, 2026

该研究解决了无长诱导路径图的着色问题，通过改进Gyárfás路径论证方法，得到了更优的着色数上界。

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Countable Graphs with Finite Path-width: Characterisation and Universality

Aug 27, 2026

研究了可数无限图的路径宽度和线宽，通过特定条件刻画有限路径宽度的图，并探讨了具有有界路径或线宽图的普遍性问题。

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Erdős-Pósa property for induced packings of long $S$-cycles

Aug 23, 2026

本文解决了长S-圈的诱导装包问题，通过引入基于脆弱耳的新耳分解技术，证明了存在多项式函数f(k, l)，使得每个图要么包含k个长度至少为l的S-圈的诱导装包，要么存在至多f(k, l)个顶点的集合与所有长度至少为l的S-圈相交。

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Where Treewidth and Pathwidth Diverge: Towards a Uniform Kernel for Pathwidth-$η$ Deletion

Aug 10, 2026

This work addresses the long-standing open question of whether the Pathwidth-η Deletion problem admits a polynomial kernel independent of η—a question that has received little attention despite the perceived similarity between treewidth and pathwidth. By integrating techniques from parameterized complexity, graph decomposition, and kernelization, the paper establishes—for the first time—the existence of uniform polynomial kernels under three distinct structural parameterizations, including vertex cover number and distance to treedepth. These results challenge the conventional intuition that treewidth and pathwidth exhibit analogous behavior in kernelization contexts. Moreover, the study conjectures that a uniform kernel also exists when parameterized solely by the solution size k, opening a new direction for future research.

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Multiway $f$-Cut is fixed-parameter tractable

Aug 10, 2026

This study addresses the Multiway f-Cut problem: given a connectivity function, a set of terminals, and an integer \(k\), determine whether there exists a partition separating all terminals such that the total cut value does not exceed \(k\). The authors leverage properties of submodular functions and parameterized algorithmic techniques to establish, for the first time, that this problem is fixed-parameter tractable with respect to the parameter \(k\), even when the connectivity function is accessible only via a value oracle. This result not only confirms the general fixed-parameter solvability of Multiway f-Cut but also significantly generalizes and simplifies known results for the classical Edge Multiway Cut problem on graphs.

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