（参数化）避免禁式的图排序问题的复杂性
The (Parameterized) Complexity of Ordering a Graph While Avoiding a Forbidden Pattern
浏览论文内容
中文总结 AI 辅助
本文研究避免禁式的图排序问题的复杂性，证明其为Σ₂^P完全且在多限制下仍难解，同时给出三类易处理性结果以构建统一图类识别框架。
中文摘要 AI 辅助
本文研究模式避免问题，即判定给定图$G$是否存在一个线性顶点序，使得该序在每个子序上都避免给定模式$P$，这里的模式是具有强制边和禁边的顶点序列。这类模式是序不变环境下诱导子图的自然有序对应，已知模式避免问题可涵盖带宽、顶点着色、队列数等多种图问题，还可扩展至奇圈横截等顶点删除问题。我们证明模式避免问题是$\boldsymbol{\textsf{$\boldsymbol{\text{Σ}_2^\textsf{P}}$}}$-完全的，且即便对模式$P$和图$G$施加多种严格限制，该问题在经典和参数化意义下仍难解。作为主要贡献，我们补充了以下易处理性结果，为识别模式可定义的图类提供统一框架：1. 以$G$的顶点完整性加$|V(P)|$为参数的固定参数算法；2. 以$G$的邻域多样性加$|E(P)|$为参数的固定参数算法；3. 对森林上的模式避免问题，几乎所有常规模式都存在多项式算法。
英文摘要
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.
