1 Introduction
Simultaneous Graph Embedding is a family of problems where one is given a set of graphs with shared vertex set and is required to produce drawings of them, each satisfying certain readability properties, so that each vertex has the same position in every . The readability property that is usually pursued is the planarity of the drawing, and a large body of research has been devoted to establish the complexity of the corresponding decision problem, or to determine whether such embeddings always exist, given the number and the types of the graphs; for a survey refer to [BKR13].
These problems have been studied both from a geometric (Geometric Simultaneous Embedding  GSE) [AGKN12, EGJPSS07] and from a topological point of view (Simultaneous Embedding with Fixed Edges  SEFE) [BR16, BCDEEIKLM07, DBLP:conf/gd/Frati06]. In particular, in GSE the edges are straightline segments, while in SEFE they are topological curves, but the edges shared between two graphs and have to be drawn in the same way in and . Unless otherwise specified, we focus on the topological setting.
We study a relaxation of the SEFE problem, where the graphs can be drawn with edge crossings. However, we prohibit certain crossing configurations in the drawings , to guarantee their readability, i.e., we require that they satisfy the conditions of a graph class in the area of beyondplanarity; see [DBLP:journals/csur/DidimoLM19] for a survey on this topic. We initiate this study with the class of quasiplanar graphs [AT07, AgarwalAPPS97, FoxPS13], by requiring that no contains three mutually crossing edges.
Definition 1 (QuaSEFE)
Given a set of graphs with shared vertex set , we say that admits a QuaSEFE if there exist quasiplanar drawings of , respectively, so that each vertex of has the same position in every and each edge shared between two graphs and is drawn in the same way in and . Further, the QuaSEFE problem asks whether an instance admits a QuaSEFE.
It may be worth mentioning that the problem of computing quasiplanar simultaneous embeddings of graph pairs has been studied in the geometric setting [DBLP:journals/cj/GiacomoDLMW15, DBLP:journals/ipl/DidimoKLOS12]. Also, simultaneous embeddings have been considered in relation to another beyondplanarity geometric graph class, namely RAC graphs [ArgyriouBKS13, DBLP:journals/jgaa/BekosDKW16, DBLP:journals/tcs/EvansLM16, DBLP:journals/jgaa/Grilli18].
We prove in Section 2 that any triple of two planar graphs and a tree admits a QuaSEFE, which also implies that any pair consisting of a planar graph^{1}^{1}1A graph is planar if it admits a drawing where each edge has at most crossings. and a planar graph admits a QuaSEFE. Recall that, for the original SEFE problem, there exist even negative instances composed of two outerplanar graphs [DBLP:conf/gd/Frati06]. Further, we investigate triples of planar graphs in which the common subgraphs have specific structural properties. Finally, we show negative results in more specialized settings in Section 3 and conclude with open problems in Section 4.
2 Sufficient Conditions for QuaSEFEs
In this section, we provide several sufficient conditions for the existence of a QuaSEFE, mainly focusing on instances composed of three planar graphs , , and . We start with a theorem relating the existence of a SEFE of two of the input graphs to the existence of a QuaSEFE of the three input graphs.
Theorem 2.1
Let , , and be planar graphs with shared vertex set . If admits a SEFE, then admits a QuaSEFE, in which the drawing of is planar.
Proof
First construct a SEFE of , and then construct a planar drawing of , whose vertices have already been placed, but whose edges have not been drawn yet, using the algorithm by Pach and Wenger [PW01].
The drawing of is planar, by construction. The drawing of is quasiplanar, as it is partitioned into two subgraphs, and , each of which is drawn planar. Analogously, the drawing of is quasiplanar.
Since every pair composed of a planar graph and a tree admits a SEFE [DBLP:conf/gd/Frati06], we derive from Theorem 2.1 the following positive result for the QuaSEFE problem.
Corollary 1
Let and be planar graphs and be a tree with shared vertex set . Then admits a QuaSEFE, in which the drawing of is planar.
Corollary 1 already shows that allowing quasiplanarity significantly enlarges the set of positive instances. We further strengthen this result, by additionally guaranteeing that even the tree is drawn planar. For this, we use a result on the partially embedded planarity [ABFJKPR15] problem (PEP): Given a planar graph , a subgraph of , and a planar embedding of , is there a planar embedding of whose restriction to coincides with ? In particular, we will exploit the following characterization, which is the core of a lineartime algorithm for PEP.
Lemma 1 ([Abfjkpr15])
Let be an instance of PEP. A planar embedding of is a solution for if and only if the following conditions hold: (C.1) for every vertex , the edges incident to in appear in the same cyclic order in the rotation schemes of in and in ; and (C.2) for every cycle of , and for every vertex of , we have that lies in the interior of in if and only if it lies in the interior of in .
Theorem 2.2
Let and be planar graphs and be a tree with shared vertex set . Then admits a QuaSEFE, in which the drawings of and are planar.
Proof
Consider planar embeddings and of and , respectively. We draw according to . This fixes the embedding of the subgraph of , thus resulting in an instance of the PEP problem. Since is acyclic, Condition C.2 of Lemma 1 is trivially fulfilled. Also, since every rotation scheme of is planar, we choose for the edges of an order compatible with , still satisfying Condition C.1. Finally, we draw the remaining edges of by considering the instance of PEP defined by its embedded subgraph . Condition C.2 is trivially satisfied, and Condition C.1 is satisfied by construction, if we add the edges of according to . Since crossings edges of the same graph belong to and , the drawing of is quasiplanar.
The additional property guaranteed by Theorem 2.2 is crucial to infer the first result in the simultaneous embedding setting for a class of beyondplanar graphs.
Theorem 2.3
Let be a planar graph and be a planar graph. Then admits a QuaSEFE.
Proof
As is planar, it is the union of a planar graph and a forest [A14]. We augment to a tree . By Theorem 2.2, there is a QuaSEFE of where and are drawn planar. Thus, is drawn quasiplanar.
We now study properties of the subgraphs induced by the edges that belong to one, to two, or to all the input graphs. We denote by the subgraph induced by the edges only in ; by the subgraph induced by the edges only in and ; and by the subgraph induced by the edges in all graphs; see Fig. (a)a.
The following two corollaries of Theorem 2.1 list sufficient conditions for and to have a SEFE. In the first case, has a unique embedding, which fulfills the conditions of Lemma 1 with respect to any planar embedding of and of . In the second case, this is because is a subgraph of .
Corollary 2
Let , , be planar graphs with shared vertex set . If is acyclic and has maximum degree , then admits a QuaSEFE.
Corollary 3
Let , , be planar graphs with shared vertex set . If , then admits a QuaSEFE.
Contrary to the previous corollaries, Theorem 2.1 has no implication for the graph , as there are instances with where no pair of graphs has a SEFE. However, we show that a simple structure of is still sufficient for a QuaSEFE.
Theorem 2.4
Let , , be planar graphs with shared vertex set . If has a planar embedding that can be extended to a planar embedding of each graph , then admits a QuaSEFE.
Proof
We draw the graph with embedding , the graph with embedding , and the graph with embedding . Then, the edges of are partitioned into two sets, one belonging to and one to , each of which is drawn planar. As the same holds for the edges of and , the statement follows.
Corollary 4
Let , , be planar graphs with shared vertex set . If is acyclic and has maximum degree , then admits a QuaSEFE.
The above discussion shows that, if one of the seven subgraphs in Fig. (a)a is empty, or has a sufficiently simple structure, admits a QuaSEFE. Most notably, this is always the case in the sunflower setting [DBLP:journals/tcs/AngeliniLN15, DBLP:journals/jgaa/HaeuplerJL13, DBLP:journals/jgaa/Schaefer13], in which every edge belongs either to a single graph or to all graphs, i.e., . We extend this result to any set of planar graphs. We remark that SEFE is NPcomplete in the sunflower setting for three planar graphs [DBLP:journals/tcs/AngeliniLN15, DBLP:journals/jgaa/Schaefer13].
Theorem 2.5
Let be planar graphs with shared vertex set in the sunflower setting. Then admits a QuaSEFE.
Proof
Let be the graph induced by the edges belonging to all graphs. We independently draw planar the graph and every subgraph , for . This guarantees that each is drawn quasiplanar.
We remark that all our proofs are constructive. Moreover, the corresponding algorithms run in linear time, as they exploit lineartime algorithms for constructing planar embeddings of graphs [DBLP:journals/jacm/HopcroftT74], for extending their partial embeddings [ABFJKPR15], and for partitioning planar graphs into planar graphs and forests [A14].
3 Counterexamples for QuaSEFE
In this section we complement our positive results, by providing negative instances of the QuaSEFE problem in two specific settings. We start with a negative result about the existence of a simple QuaSEFE for two quasiplanar graphs and one star. Here simple means that a pair of independent edges in the same graph is allowed to cross at most once and a pair of adjacent edges in the same graph is not allowed to cross. Note that our algorithms in Section 2 may produce nonsimple drawings. Also, the maximum number of edges in a quasiplanar graph with vertices depends on whether simplicity is required or not [AT07].
Theorem 3.1
There exist two quasiplanar graphs , and a star with shared vertex set such that does not admit a simple QuaSEFE.
Proof
Let and let be the edges of the complete graph on . Further, let , let , and let . By construction, is the star on all eleven vertices with center , while Fig. (b)b shows that there is a simple quasiplanar drawing of (and of , which is a subgraph of , up to vertex relabeling).
Suppose that has a simple QuaSEFE, and let be the drawing of the union of and that is part of it. Since the union of and has edges, which exceeds the upper bound of edges in a simple quasiplanar graph [AT07], is not simple or not quasiplanar. Since is the only edge in that is not in , edge is involved in every crossing violating simplicity or quasiplanarity. Analogously, one of , say , is involved in every crossing violating simplicity or quasiplanarity; in particular, crosses . Since both and belong to , the drawing of that is part of the simple QuaSEFE is not simple, a contradiction.
The second special setting is the one in which one of the input graphs is already drawn in a quasiplanar way, and the goal is to draw the other input graphs so that the resulting simultaneous drawing is a QuaSEFE. This setting is motivated by the natural approach, for an instance , of first constructing a solution for and then adding the remaining edges of . Note that, since the drawing of the first graph partially fixes a drawing of the second graph, this can be seen as a version of the PEP problem for quasiplanarity.
For the original SEFE problem, this setting always has a solution when the graph that is already drawn (in a planar way) is a general planar graph, and the other graph is a tree [DBLP:conf/gd/Frati06]. In a surprising contrast, we construct negative instances for the QuaSEFE problem that are composed of two matchings only.
Theorem 3.2
Let and be two matchings on the same vertex set and let be a quasiplanar drawing of . Instance does not always admit a QuaSEFE in which the drawing of is .
Proof
First recall that the edges in have to be drawn in the quasiplanar drawing of as they are in . Consider the quasiplanar drawing of the matching , with , in Fig. (c)c, and let contain the edges and . Since is enclosed in a region bounded by the crossing edges and , in any quasiplanar drawing of edge crosses exactly one of and . In the first case, crosses also and (dotted blue). In the second case, crosses also and (dashed red). In both cases, and cannot be crossed, and thus cannot be drawn so that is quasiplanar.
4 Conclusions and Open Problems
We initiated the study of simultaneous embeddability in the beyond planar setting, which is a fertile and almost unexplored research direction that promises to significantly enlarge the families of representable graphs when compared with the planar setting. We conclude the paper by listing a few open problems.

A natural question is whether two planar graphs, a quasiplanar graph and a matching, three outerplanar graphs, or four paths admit a QuaSEFE. All our algorithms construct drawings with a stronger property than quasiplanarity, namely that they are composed of two sets of planar edges. Exploiting quasiplanarity in full generality may lead to further positive results.

Another intriguing direction is to determine the computational complexity of the QuaSEFE problem, both in its general version and in the two restrictions studied in Section 3. In particular, the setting in which one of the graphs is already drawn can be considered as a quasiplanar version of the PEP problem, which is known to be lineartime solvable in the planar case [ABFJKPR15].

Extend the study to other beyondplanarity classes. For example, do any two planar graphs admit a planar SEFE for some constant ?