A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. It is maximal 1-planar if the addition of any edge violates 1-planarity. Maximal 1-planar graphs have at most 4n − 8 edges. We show that there are sparse maximal 1-planar graphs with only 45 17 n + O(1) edges. With a fixed rotation system there are maximal 1-planar graphs with only 7 3 n+O(1) edges. This is sparser than maximal planar graphs. There cannot be maximal 1-planar graphs with less than 21 10 n − O(1) edges and less than 28 13 n − O(1) edges with a fixed rotation system. Furthermore, we prove that a maximal 1-planar rotation system of a graph uniquely determines its 1-planar embedding.

A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. 1-planarity is known NP-hard, even for graphs of bounded bandwidth, pathwidth, or treewidth, and for near-planar graphs in which an edge is added to a planar graph. On the other hand, there is a linear time 1-planarity testing algorithm for maximal 1-planar graphs with a given rotation system. In this work, we show that 1-planarity remains NP-hard even for 3-connected graphs with (or without) a rotation system. Moreover, the crossing number problem remains NP-hard for 3-connected 1-planar graphs with (or without) a rotation system.

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Abstract. In stack and queue layouts the vertices of a graph are linearly ordered from left to right, where each edge corresponds to an item and the left and right end vertex of each edge represents the addition and removal of the item to the used data structure. A graph admitting a stack or queue layout is a stack or queue graph, respectively.Typical stack and queue layouts are rainbows and twists visualizing the LIFO and FIFO principles, respectively. However, in such visualizations, twists cause many crossings, which make the drawings incomprehensible. We introduce linear cylindric layouts as a visualization technique for queue and deque (double-ended queue) graphs. It provides new insights into the characteristics of these fundamental data structures and extends to the visualization of mixed layouts with stacks and queues. Our main result states that a graph is a deque graph if and only if it has a plane linear cylindric drawing.

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