Simultaneous Embedding of Planar Graphs

Abstract

Simultaneous embedding is concerned with simultaneously representing a series of graphs sharing some or all vertices. This forms the basis for the visualization of dynamic graphs and thus is an important field of research. Recently there has been a great deal of work investigating simultaneous embedding problems both from a theoretical and a practical point of view. We survey recent work on this topic.

Synchronized Planarity with Applications to Constrained Planarity Problems

“…This makes Thickenability a special case of Synchronized Planarity. Via the reduction from Connected SEFE to Clustered Planarity given by Angelini and Da Lozzo [2], the above result extends to Connected SEFE, which was a major open problem in the context of simultaneous graph representations [8]. We flatten this chain of reductions by giving a simple linear reduction from each of the problems Connected SEFE, Clustered Planarity, and Atomic Embeddability to Synchronized Planarity, yielding quadratic time algorithms for all of them.…”

Synchronized Planarity with Applications to Constrained Planarity Problems

“…The last reduction to Thickenability is based on a combinatorial characterization of Thickenability by Neuwirth [23], which basically states that multiple graphs have to be embedded consistently, that is, such that the rotation is synchronized between certain vertex pairs of different graphs. Via the reduction from Connected SEFE to Clustered Planarity given by Angelini and Da Lozzo [2], the above result extends to Connected SEFE, which was a major open problem in the context of simultaneous graph representations [7]. We flatten this chain of reductions by giving a simple linear reduction from each of the problems Connected SEFE, Clustered Planarity, and Atomic Embeddability to Synchronized Planarity, yielding quadratic-time algorithms for all of them.…”