Straight-line drawings of 1-planar graphs
Copyright policy )Straight-line drawings of 1-planar graphs
A graph is 1-planar if it can be drawn in the plane so that each edge is crossed at most once. However, there are 1-planar graphs which do not admit a straight-line 1-planar drawing. We show that every 1-planar graph has a straight-line drawing with a two-coloring of the edges, so that edges of the same color do not cross. Hence, 1-planar graphs have geometric thickness two. In addition, each edge is crossed by edges with a common vertex if it is crossed more than twice. The drawings use high precision arithmetic with numbers with O(n log n) digits and can be computed in linear time from a 1-planar drawing
Computational Geometry (cs.CG), FOS: Computer and information sciences, Graph representations (geometric and intersection representations, etc.), 68R10, shift method, Planar graphs; geometric and topological aspects of graph theory, graph drawing, Graph algorithms (graph-theoretic aspects), straight-line drawings, Computer Science - Computational Geometry, geometric thickness, F.2.2, 1-planar graphs
Computational Geometry (cs.CG), FOS: Computer and information sciences, Graph representations (geometric and intersection representations, etc.), 68R10, shift method, Planar graphs; geometric and topological aspects of graph theory, graph drawing, Graph algorithms (graph-theoretic aspects), straight-line drawings, Computer Science - Computational Geometry, geometric thickness, F.2.2, 1-planar graphs
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