# Characterizations of Deque and Queue Graphs

• Christopher Auer
• Andreas Gleißner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6986)

## Abstract

In graph layouts the vertices of a graph are processed according to a linear order and the edges correspond to items in a data structure inserted and removed at their end vertices. Graph layouts characterize interesting classes of planar graphs: A graph G is a stack graph if and only if G is outerplanar, and a graph is a 2-stack graph if and only if it is a subgraph of a planar graph with a Hamiltonian cycle [2]. Heath and Rosenberg [12] characterized all queue graphs as the arched leveled-planar graphs. In [1], we have introduced linear cylindric drawings (LCDs) to study graph layouts in the double-ended queue (deque) and have shown that G is a deque graph if and only if it permits a plane LCD.

In this paper, we show that a graph is a deque graph if and only if it is the subgraph of a planar graph with a Hamiltonian path. In consequence, we obtain that the dual of an embedded queue graph contains a Eulerian path. We also turn to the respective decision problem of deque graphs and show that it is $$\mathcal{NP}$$-hard by proving that the Hamiltonian path problem in maximal planar graphs is $$\mathcal{NP}$$-hard. Heath and Rosenberg state [12] that queue graphs are “almost” proper leveled-planar. We show that bipartiteness captures this “almost”: A graph is proper leveled-planar if and only if it is a bipartite queue graph.

## Keywords

Planar Graph Hamiltonian Cycle Front Line Hamiltonian Path Outerplanar Graph
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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