Simultaneous Planar Embeddability

• Simultaneous planar embeddability is defined as the challenge of drawing multiple planar graphs with a shared vertex set while maintaining planarity and specific geometric or combinatorial constraints.
• The topic involves diverse models—GSE, SEFE, and CSE—that use different methods such as SPQR- and PQ-tree techniques to address algorithmic and complexity issues.
• It underpins applications in comparative visualization and graph morphing, with research focusing on NP-hard cases, fixed-parameter tractability, and universality in geometric graph theory.

Simultaneous Planar Embeddability investigates whether multiple planar graphs—defined on a shared set of vertices—can be drawn in the plane such that various geometric or combinatorial constraints are respected in all drawings, typically requiring planarity for each individual graph and prescribed alignments or mappings for vertices and/or edges. This area is central to comparative visualization, graph morphing, and dynamic graph drawing, and it provides challenging algorithmic and combinatorial problems with links to geometric graph theory, parameterized complexity, and the existential theory of the reals.

1. Models and Definitions

At the core of simultaneous planar embeddability are several distinct but interrelated models, each imposing different constraints:

• Geometric Simultaneous Embedding (GSE): Requires that all graphs share the same point set for common vertices and represent every graph with a straight-line, planar drawing. The mapping from vertices to points is fixed for common vertices across graphs.
• Simultaneous Embedding with Fixed Edges (SEFE): Allows edges to be drawn as arbitrary Jordan curves (not necessarily straight lines) but demands that each shared vertex be mapped to the same point and each shared edge be drawn identically in all graphs.
• Combinatorial Simultaneous Embedding (CSE): Relaxed further, only insists that the common subgraph induces the same combinatorial embedding (cyclic edge orders) in each graph; the geometric realization may vary per graph.
• Simultaneous Geometric Embedding with Mapping (k-SGE): For $k$ graphs $G_1, \ldots, G_k$ on the same $n$-vertex set $V$, asks for a set $P$ of $n$ points and a bijection $\phi: V \to P$ so that each $G_i$ admits a crossing-free straight-line drawing under $\phi$. This model is central to queries of universality, existence, and complexity (Bläsius et al., 2012, Cardinal et al., 2013).

The degree of constraint (from CSE through SEFE to GSE) directly impacts tractability, complexity, and geometric realizability.

2. Structural and Algorithmic Results

Existence and Characterization

• Straight-Line Embeddability: Not all pairs $(G_1, G_2)$ admit a geometric simultaneous embedding—even for simple classes such as a path and a planar graph—due to conflicting face or embedding constraints (Bläsius et al., 2012).
• SEFE Characterization: Jünger & Schulz established that $(G_1, G_2)$ admit a SEFE if and only if there exist planar combinatorial embeddings of $G_1$ and $G_2$ restricting to the same embedding for the common subgraph (Bläsius et al., 2012, Bläsius et al., 2015). This reduces SEFE to a question about cyclic ordering and face assignment on the common graph.

Efficient Algorithms and Complexity

• NP-Hardness and Tractable Cases: Deciding GSE is NP-hard for two graphs (Bläsius et al., 2012). SEFE is NP-complete for $k \geq 3$ (Fink et al., 2023) and remains of unknown complexity for $k=2$ general graphs (Bläsius et al., 2015). However, there are efficient algorithms (linear- to quadratic-time) for SEFE when the common subgraph is biconnected, a tree, or a pseudoforest (Bläsius et al., 2012, Bläsius et al., 2011).
• SPQR- and PQ-Tree-Based Methods: Core algorithms for SEFE leverage SPQR-trees for biconnected components and PQ-trees for encoding permissible edge orders. For biconnected $G_1, G_2$ with a connected intersection, the problem reduces to a 2-fixed Simultaneous PQ-Ordering instance, solvable in $O(n^2)$ time in general and $O(n)$ time if the maximum degree is bounded (Bläsius et al., 2011).

Parameterized Complexity

Key advances establish that SEFE is fixed-parameter tractable (FPT) with respect to various combinations of parameters on the union and shared graphs, including vertex cover number, feedback edge set number, number of cutvertices, and the maximum degree (Fink et al., 2023).

Parameterization	Result	Reference
$k + vc(G^\cup)$	FPT	(Fink et al., 2023)
$k + fes(G^\cup)$	FPT	(Fink et al., 2023)
$vc(G) + \rho(G)$	FPT	(Fink et al., 2023)
$cv(G) + \Delta(G)$	FPT	(Fink et al., 2023)
singleton parameters	NP-hard	(Fink et al., 2023)

3. Universality, Conflict Collections, and Lower Bounds

A fundamental question is whether universal point sets exist for all planar graphs of a given order $n$, and how large such sets must be.

• Universal Point Sets: The minimal size $f_p(n)$ for $n$-universal point sets is bounded as $n \leq f_p(n) \leq n^2/4 + \Theta(n)$ (Goenka et al., 2023). For stacked triangulations, $f_s(n) \geq (\alpha-o(1)) n$ with $\alpha\approx 1.293$ (Scheucher et al., 2018).
• Conflict Collections: These are sets of $n$-vertex planar graphs that cannot all be simultaneously embedded on any $n$-point set. Initially, only superexponential upper bounds were known ($O(n\,n!)$), but recent progress yields $O(n\cdot 4^{n/11})$ (Goenka et al., 2023), and by probabilistic methods, as few as $(3+o(1))\log_2 n$ graphs for sufficiently large $n$ (Steiner, 2023). Explicit constructions reach $O(n^6)$, while computer verification gives 49-graph conflict collections for $n=11$ (Scheucher et al., 2018).

4. Bends, Crossings, and Polyline Embeddings

Where exact geometric simultaneous drawing is impossible, polyline drawings with a bounded number of bends per edge become the measure of quality.

• Poly-Bend SEFE: Any two planar graphs admit a simultaneous embedding with at most three bends per edge on an $O(n^2)\times O(n^2)$ grid (Bläsius et al., 2012). Recent refinements show that for uniformly random planar graphs, $O(n^{1-1/k})$ bends per edge are both achievable and optimal for $k$ graphs with fixed vertex mappings (Gordon, 2012).
• SEFE with Few Bends and Crossings: For two trees, one bend per edge and at most four crossings per edge pair. For two planar graphs, six bends per edge and sixteen crossings suffice (Frati et al., 2015).
• Extension to Colored Graphs and Universal Point Sets: The polyline bend complexity can be kept sublinear in $n$ for compatibly colored graphs, and universal point sets for colored graphs give $O(b)$ bends for $n\lceil c/b\rceil$ locations (Mondal, 2021).

5. Generalizations, Structural Decompositions, and PQ-Ordering

• Simultaneous PQ-Ordering: Introduced as a meta-problem which unifies several simultaneous embedding questions. NP-complete in general, polynomial-time solvable when the critical structure ("2-fixed instances") is bounded (Bläsius et al., 2011). The reduction of SEFE for biconnected graphs with connected intersection to 2-fixed Simultaneous PQ-Ordering yields a uniform algebraic framework and efficient algorithms.
• Block and Cutvertex Decomposition: SEFE algorithms preprocess by removing union cutvertices, union-separating pairs, and replacing general biconnected components with cycles, facilitated by SPQR- and PQ-tree machinery (Bläsius et al., 2015).

6. Disconnected Intersections, Relative Positions, and the CC-tree

When the common graph is disconnected, modeling the relative locations of components becomes essential:

• CC-tree and CC$^\oplus$-tree: Data structures used to encode all possible embeddings of disconnected cycle unions and general connected components with fixed embeddings, respectively. The intersection algorithm for CC-trees enables linear-time solutions when the common graph is a set of cycles and quadratic-time solutions when components have fixed embeddings (Bläsius et al., 2012).
• Relative Position Constraints: The inclusion of cyclic and face-based constraints for components in disconnected common graphs is necessary for generalized SEFE algorithms.

7. Complexity Frontiers and Future Directions

• Existential Theory of the Reals ($\exists\mathbb{R}$): The $k$-SGE problem is $\exists\mathbb{R}$-complete, i.e., as hard as deciding the truth of existential sentences over the reals. Thus, while many instances are efficiently decidable, the general problem for two or more graphs remains polynomially equivalent to highly nontrivial geometric computing questions like pseudoline stretchability. Any explicit embedding might require doubly-exponential precision in the worst case (Cardinal et al., 2013).
• Open Problems: The value of $\sigma(n)$ for small $n$ ("is there a conflict collection of size 2"), tight lower/upper bounds for universal point sets, polynomial-time recognition for SEFE with general intersections, and minimally restricted, efficiently-solvable instances remain at the research forefront. Unifying data structures extending PQ-trees that can efficiently handle multiple cutvertices and multiblock constraints is a promising direction (Bläsius et al., 2011, Bläsius et al., 2012).

Simultaneous planar embeddability thus lies at the intersection of combinatorial topology, computational geometry, and parameterized complexity, combining structural graph theory with geometric realization and algorithm design. The subject’s ongoing development continues to reveal rich combinatorial structure and surprising algorithmic phenomena, shaped by increasingly deep connections to universality, order types, and hardness in geometric computation.