Abstract
We examine the simultaneous embedding with xed edges problem for two planar graphs G1 and G2 with the focus on their intersection S = G1\G2. In particular, we will present the complete set of intersection graphs S that guarantee a simultaneous embedding with xed edges for (G1;G2). More formally, we dene the subset ISEFE of all planar graphs as follows: A graph S lies in ISEFE if every pair of planar graphs (G1;G2) with intersection S = G1 G2 has a simultaneous embedding with xed edges. We will characterize this set by a detailed study of topological embeddings and nally give a complete list of graphs in this set as our main result of this paper.
Highlights
A simultaneous embedding with fixed edges (SEFE) of two graphs G1 and G2 is a pair of drawings D1 of G1 and D2 of G2 such that each drawing is planar and the intersection S = G1 ∩ G2 is drawn in both drawings
Di Giacomo and Liotta [2] extended this result by showing that any pair of an outerplanar graph with a cycle has a simultaneous embedding with fixed edges while Frati [6] showed that any pair of a planar graph and a tree has a simultaneous embedding with fixed edges
We define the subset ISEFE of all planar graphs as follows: A graph S lies in ISEFE if every pair of planar graphs (G1, G2) with intersection S = G1 ∩ G2 has a simultaneous embedding with fixed edges
Summary
A simultaneous embedding with fixed edges (SEFE) of two graphs G1 and G2 is a pair of drawings D1 of G1 and D2 of G2 such that each drawing is planar and the intersection S = G1 ∩ G2 is drawn in both drawings. The problem to decide whether a graph pair has a simultaneous embedding with fixed edges or not has been studied from different angles. Restrict G1 and/or G2 to certain classes of planar graphs and make a statement whether any pair of these graph types has a simultaneous embedding with fixed edges or not. In this paper we examine the simultaneous embedding with fixed edges problem for two planar graphs G1 and G2 from a different point of view. Rather than forcing G1 or G2 to be a specific graph we examine which types of intersections allow a simultaneous embedding with fixed edges for general graphs G1 and G2. This condition is irrelevant for most of our examinations but leads to a nice formulation of our main result as it is described in Theorem 5
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