Simultaneous Embedding of Colored Graphs

Abstract

A set of colored graphs are compatible, if for every color i, the number of vertices of color i is the same in every graph. A simultaneous embedding of k compatibly colored graphs, each with n vertices, consists of k planar polyline drawings of these graphs such that the vertices of the same color are mapped to a common set of vertex locations.We prove that simultaneous embedding of k ∈ o(log log n) colored planar graphs, each with n vertices, can always be computed with a sublinear number of bends per edge. Specifically, we show an O(min{c, n 1−1/γ }) upper bound on the number of bends per edge, where γ = 2 k/2 and c is the total number of colors. Our bound, which results from a better analysis of a previously known algorithm [Durocher and Mondal, SIAM J. Discrete Math., 32(4), 2018], improves the bound for k, as well as the bend complexity by a factor of √ 2 k . The algorithm can be generalized to obtain small universal point sets for colored graphs. We prove that n c/b vertex locations, where b ≥ 1, suffice to embed any set of compatibly colored n-vertex planar graphs with bend complexity O(b), where c is the number of colors.