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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.

Abstract.A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. In general, 1-planar graphs do not admit straightline drawings. We show that every 3-connected 1-planar graph has a straight-line drawing on an integer grid of quadratic size, with the exception of a single edge on the outer face that has one bend. The drawing can be computed in linear time from any given 1-planar embedding of the graph.

Abstract. In this manuscript, we present several challenging and interesting open problems in graph drawing. The goal of the listing in this paper is to stimulate future research in graph drawing.

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