The (Parameterized) Complexity of Ordering a Graph While Avoiding a Forbidden Pattern

2026-08-31

AI Summary

本文研究了图排序中避免给定模式的问题，提出了基于图的顶点完整性、邻域多样性和森林结构的多项式或固定参数算法。

Abstract

In this paper, we study the Pattern Avoidance problem of determining whether a given graph $G$ admits a linear vertex order which avoids a given pattern $P$, i.e., a vertex sequence with some forced and forbidden edges, on every suborder. Such patterns form a natural ordered counterpart to induced subgraphs in the order-invariant setting, and it is known that Pattern Avoidance captures a broad variety of graph problems including Bandwidth, Vertex Coloring, Queue Number, and extends to vertex-deletion problems such as Odd Cycle Transversal. We show that Pattern Avoidance is $Σ_2^{\textsf{P}}$-complete and furthermore remains intractable (in both the classical and parameterized sense) even under a variety of severe restrictions to both the pattern $P$ and the graph $G$. As our main contributions, we complement these lower bounds with the following tractability results, which provide a unifying framework for recognizing pattern-definable graph classes: - a fixed-parameter algorithm w.r.t. the vertex integrity of $G$ plus $|V(P)|$, - a fixed-parameter algorithm w.r.t. the neighborhood diversity of $G$ plus $|E(P)|$, and - a polynomial algorithm for Pattern Avoidance on forests for almost all constant-sized patterns.

Problem

• Pattern Avoidance
• graph problems
• linear vertex order
• forbidden pattern
• induced subgraphs

Innovation

• fixed-parameter algorithm
• vertex integrity
• neighborhood diversity
• polynomial algorithm
• pattern-definable graph classes