Characterizations of Deque and Queue Graphs

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.

Supported by the Deutsche Forschungsgemeinschaft (DFG), grant Br835/15-1.