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 $k\in o(\log \log n)$ colored planar graphs, each with n vertices, can always be computed with a sublinear number of bends per edge. Specifically, we show an $O(\min \{c, n^{1-1/\gamma }\})$ upper bound on the number of bends per edge, where $\gamma = 2^{\lceil k/2 \rceil }$ and c is the total number of colors. Our bound, which results from a better analysis of a previously known algorithm by Durocher and Mondal (SIAM J Discrete Math 32(4):2703–2719, 2018), improves the bound for k, as well as the bend complexity by a factor of $\sqrt{2}^{k}$. The algorithm can be generalized to obtain small universal point sets for colored graphs. We prove that $n\lceil c/b \rceil$ vertex locations, where $b\ge 1$, suffice to embed any set of compatibly colored n-vertex planar graphs with bend complexity O(b), where c is the number of colors.