Simultaneous Embedding of Colored Graphs

TL;DR

This paper presents improved algorithms for simultaneously embedding multiple compatibly colored planar graphs with a sublinear number of bends per edge, advancing the understanding of graph visualization with color constraints.

Contribution

It introduces a refined analysis and bounds for simultaneous embeddings of colored graphs, reducing the bend complexity and extending the applicability to universal point sets.

Findings

Simultaneous embedding of up to o(log log n) colored graphs with sublinear bends is always possible.

Provides an $O( ext{min}igrace c, n^{1-1/ ext{ extgamma}}igrace)$ bound on bends per edge.

Improves previous bounds by a factor of $ extsqrt{2}^k$ for the number of 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 [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 $\sqrt{2}^{k}$. 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.