AI 中文总结
研究图的栈和队列布局问题，引入k缺陷栈和队列布局的松弛概念，允许同一集合内边有一定交叉或嵌套关系，从组合和算法角度研究，给出不同图类和参数下关于有缺陷线性布局的一系列结果。
AI 中文摘要
图的线性布局，特别是栈和队列布局，在图绘制中是成熟的表示类型。在栈布局中，同一集合内无两条独立边交叉；队列布局中，同一集合内无两条独立边嵌套。核心问题是确定给定图的栈数或队列数。本文引入有缺陷的栈和队列布局，对于给定整数k>0，k缺陷栈布局（或队列布局）允许一条边与同一集合内至多k条边有交叉（或嵌套）关系。我们从组合和算法角度研究有缺陷的线性布局，给出不同图类和参数的一系列结果。
英文摘要
Linear layouts of graphs -- particularly \emph{stack} and \emph{queue} layouts -- are well-established types of representations in graph drawing, thanks to their connection with numerous theoretical and practical problems. In such layouts, all vertices are linearly ordered and the edges are partitioned into sets that avoid specific forbidden configurations: in a stack layout no two independent edges within the same set cross, whereas in a queue layout no two independent edges within the same set are nested. A central problem in this context is to determine, for a given graph $G$, its \emph{stack number} or \emph{queue number}, that is, the minimum number of sets into which the edges can be partitioned so that a corresponding stack or queue layout of $G$ exists. In this work, we introduce a relaxation of stack and queue layouts, which allows some forbidden patterns for the edges in the same set. Namely, for a given integer $k > 0$, a \emph{$k$-defective stack layout} (resp. a \emph{$k$-defective queue layout}) allows an edge to be in a crossing (resp. nesting) relationship with at most~$k$ edges within the same set. Our motivation is to extend the classes of graphs that admit linear layouts using a limited number of edge-partition sets, at the cost of allowing some defects. We study defective linear layouts both from a combinatorial and from an algorithmic perspective, providing an array of results across different graph classes and parameters.
Comments38 pages, 22 figures
