On a Tree and a Path with no Geometric Simultaneous Embedding

Abstract

Two graphs G1 = (V, E1) and G2 = (V, E2) admit a geometric simultaneous embedding if there exist a set of points P and a bijection M : V → P that induce planar straight-line embeddings both for G1 and for G2. The most prominent problem in this area is the question of whether a tree and a path can always be simultaneously embedded. We answer this question in the negative by providing a counterexample. Additionally, since the counterexample uses disjoint edge sets for the two graphs, we also negatively answer another open question, that is, whether it is possible to simultaneously embed two edge-disjoint trees. Finally, we study the same problem when some constraints on the tree are imposed. Namely, we show that a tree of height 2 and a path always admit a geometric simultaneous embedding. In fact, such a strong constraint is not so far from closing the gap with the instances not admitting any solution, as the tree used in our counterexample has height 4.

A Survey on Graph Drawing Beyond Planarity

“…For example, Di Giacomo et al study simultaneous geometric quasi planar embedding (SGQPE), where each Γ i is a straight-line quasi-planar drawing [95]. They show for instance that a tree and a path always admit an SGQPE, in contrast with the negative result in the simultaneous geometric planar embedding setting [20]. More in general, they prove that trees and other meaningful subfamilies of the outerplanar graphs admit an SGQPE.…”

The QuaSEFE Problem

“…These problems have been studied both from a geometric (Geometric Simultaneous Embedding -GSE) [6,16] and from a topological point of view (Simultaneous Embedding with Fixed Edges -SEFE) [10,12,19]. In particular, in GSE the edges are straight-line segments, while in SEFE they are topological curves, but the edges shared between two graphs G i and G j have to be drawn in the same way in Γ i and Γ j .…”

On the NP-hardness of GRacSim drawing and k-SEFE Problems

“…The problem of computing a simultaneous embedding of two or more graphs has been extensively explored by the graph drawing community. Indeed, besides its inherent theoretical interest [1,2,4,5,6,7,9,10,11,12,13,14,15,16,17,18,19,22,23,24,25,26], it has several applications in dynamic network visualization, especially when a visual analysis of an evolving network is needed. Although many variants of this problem have been investigated so far, a general formulation for two graphs can be stated as follows: Let G 1 = (V 1 , E 1 ) and G 2 = (V 2 , E 2 ) be two planar graphs sharing a common (or shared) subgraph G = (V, E), where V = V 1 ∩ V 2 and E = E 1 ∩ E 2 .…”

“…The SGE problem is therefore a restricted version of SEFE, and it turned out to be "too much restrictive", i.e. there are examples of pairs of structurally simple graphs, such as a path and a tree [6], that do not admit an SGE. Also, testing whether two planar graphs admit a simultaneous geometric embedding is N P-hard [16].…”