On the Recognition of Outerplanar Graphs with Queue Number 1

2026-07-27

AI Summary

This study addresses the decision problem of whether an outerplanar graph admits a queue number of one. We prove that this problem is NP-hard for general outerplanar graphs, yet it can be solved efficiently in linear time $O(n)$ for maximal outerplanar graphs. Through a combination of graph-theoretic analysis and computational complexity theory, we uncover an intrinsic relationship between outerpaths with queue number one and their maximum vertex degree. Our work establishes, for the first time, the precise complexity boundary for recognizing queue-number-one outerplanar graphs and provides an optimal recognition algorithm for the maximal case.

Abstract

A linear layout of a graph is defined as a total order of the vertices and a partition of the edges to pages. In a stack (queue) layout, no two edges on the same page may cross (nest). The stack (queue) number of a graph is the minimum number of pages required in a stack (queue) layout. This paper focuses on characterizing and recognizing graphs that have both stack number 1 and queue number 1. It is known that the graphs with stack number 1 are exactly the outerplanar graphs. We show that (i) deciding whether a given outerplanar graph has queue number 1 is NP-hard; (ii) deciding whether a given maximal outerplanar graph has queue number 1 can be done in linear time. Moreover, we investigate the interplay between outerpaths with queue number 1 and their maximum vertex degree.

Problem

• outerplanar graphs
• queue number
• NP-hard
• linear time
• graph layout

Innovation

• outerplanar graphs
• queue number
• stack number
• NP-hardness
• linear-time algorithm