Constrained Simultaneous and Near-Simultaneous Embeddings

Abstract

Abstract.A geometric simultaneous embedding of two graphs G1 = (V1, E1) and G2 = (V2, E2) with a bijective mapping of their vertex sets γ : V1 → V2 is a pair of planar straight-line drawings Γ1 of G1 and Γ2 of G2, such that each vertex v2 = γ(v1) is mapped in Γ2 to the same point where v1 is mapped in Γ1, where v1 ∈ V1 and v2 ∈ V2.In this paper we examine several constrained versions and a relaxed version of the geometric simultaneous embedding problem. We show that if the input graphs are assumed to share no common edges this does not seem to yield large classes of graphs that can be simultaneously embedded. Further, if a prescribed combinatorial embedding for each input graph must be preserved, then we can answer some of the problems that are still open for geometric simultaneous embedding. Finally, we present some positive and negative results on the near-simultaneous embedding problem, in which vertices are not mapped exactly to the same but to "near" points in the different drawings.

Geometric RAC Simultaneous Drawings of Graphs

“…Cabello et al [7] showed that a geometric simultaneous drawing of a matching and (a) a wheel, (b) an outerpath or, (c) a tree always exists, while there exist a planar graph and a matching that cannot be drawn simultaneously. For a quick overview of known results refer to Table 1 of [14].…”

Simultaneous Embeddings with Few Bends and Crossings

“…A related interesting problem is to determine how many bends per edge are needed to construct a simultaneous embedding (without fixed edges) of pairs of (simple) planar graphs. The best known upper bound is two [8,9,19] and the best known lower bound is one [15]. As a final research direction, we mention the problem of constructing SEFEs of pairs of planar graphs in polynomial area, while matching our bounds for the number of bends and crossings.…”

On a Tree and a Path with no Geometric Simultaneous Embedding

“…Hence, vertex v 4 has to be inside Q 3 , as otherwise edge (v 1 , v 4 ) would cross one of p 1 or p 2 . However, in this case, there is no placement for vertices v 7 and v 10 that avoids a crossing between one of edges (v 4 , v 7 ) or (v 4 , v 10 ) and one of the already drawn edges.…”

“…The research on this problem opened a new exciting field of problems and techniques, like ULP trees and graphs [5,7,8], colored simultaneous embedding [1], nearsimultaneous embedding [10], and matched drawings [3], deeply related to the general fundamental question of point-set embeddability.…”