Stack and Queue Layouts with Defects

2026-07-22

AI Summary

Traditional stack and queue layouts require edge sets to be entirely non-crossing or non-nested, which severely restricts the classes of graphs they can handle. This work proposes a *k*-defective stack/queue layout model that permits each edge to cross (in stacks) or nest (in queues) with at most *k* other edges within the same set, thereby substantially relaxing these constraints while preserving the linear layout structure. Through combinatorial graph-theoretic analysis and algorithmic design, the study systematically investigates the existence, upper and lower bounds, and constructive methods for such layouts across various graph classes. The results extend the theory of linear layouts and establish, for the first time, a quantitative relationship between the complexity and expressive power of defective layouts.

Abstract

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.

Problem

• stack layout
• queue layout
• defective layout
• graph drawing
• linear layout

Innovation

• defective layouts
• stack number
• queue number
• graph drawing
• linear layouts