Abstract
A graph layout describes the processing of a graph G by a data structure \(\mathcal{D}\), and the graph is called a \(\mathcal{D}\)-graph. The vertices of G are totally ordered in a linear layout and the edges are stored and organized in \(\mathcal{D}\). At each vertex, all edges to predecessors in the linear layout are removed and all edges to successors are inserted. There are intriguing relationships between well-known data structures and classes of planar graphs: The stack graphs are the outerplanar graphs [4], the queue graphs are the arched leveled-planar graphs [12], the 2-stack graphs are the subgraphs of planar graphs with a Hamilton cycle [4], and the deque graphs are the subgraphs of planar graphs with a Hamilton path [2]. All of these are proper subclasses of the planar graphs, even for maximal planar graphs.
We introduce splittable deques as a data structure to capture planarity. A splittable deque is a deque which can be split into sub-deques. The splittable deque provides a new insight into planarity testing by a game on switching trains. Here, we use it for a linear-time planarity test of a given rotation system.
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Auer, C., Brandenburg, F.J., Gleißner, A., Hanauer, K. (2013). Characterizing Planarity by the Splittable Deque. In: Wismath, S., Wolff, A. (eds) Graph Drawing. GD 2013. Lecture Notes in Computer Science, vol 8242. Springer, Cham. https://doi.org/10.1007/978-3-319-03841-4_3
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