The Rique-Number of Graphs

TL;DR

This paper investigates linear graph layouts constrained by a specialized queue structure called rique, characterizes graphs with single-page rique layouts, and explores the rique-number for complete graphs.

Contribution

It introduces the concept of rique layouts, characterizes graphs with single-page layouts, and provides bounds on the rique-number for complete graphs.

Findings

Characterization of graphs with single-page rique layouts.

Development of a testing algorithm for maximal planar graphs.

Bounds established for the rique-number of complete graphs.

Abstract

We continue the study of linear layouts of graphs in relation to known data structures. At a high level, given a data structure, the goal is to find a linear order of the vertices of the graph and a partition of its edges into pages, such that the edges in each page follow the restriction of the given data structure in the underlying order. In this regard, the most notable representatives are the stack and queue layouts, while there exists some work also for deques. In this paper, we study linear layouts of graphs that follow the restriction of a restricted-input queue (rique), in which insertions occur only at the head, and removals occur both at the head and the tail. We characterize the graphs admitting rique layouts with a single page and we use the characterization to derive a corresponding testing algorithm when the input graph is maximal planar. We finally give bounds on the number of needed pages (so-called rique-number) of complete graphs.