Testing Simultaneous Planarity when the Common Graph is 2-Connected

Two planar graphs G1 and G2 sharing some vertices and edges are `simultaneously planar’ if they have planar drawings such that a shared vertex [edge] is represented by the same point [curve] in both drawings. It is an open problem whether simultaneous planarity can be tested efficiently. We give a linear-time algorithm to test simultaneous planarity when the two graphs share a 2-connected subgraph. Our algorithm extends to the case of k planar graphs where each vertex [edge] is either common to all graphs or belongs to exactly one of them.

💡 Research Summary

The paper addresses the long‑standing open problem of efficiently testing simultaneous planarity when two planar graphs share a non‑trivial subgraph. Simultaneous planarity asks whether two (or more) planar graphs can be drawn in the plane such that every vertex and edge that belongs to both graphs is represented by the same point or curve in both drawings. While polynomial‑time algorithms exist when the common part is a tree or merely connected, the case where the shared subgraph is 2‑connected has resisted a fast solution.

The authors present a linear‑time algorithm (O(|V|+|E|)) for the situation where the common subgraph C is 2‑connected. The core of the method relies on the SPQR‑tree decomposition of C, a classic data structure that captures all possible planar embeddings of a 2‑connected graph. An SPQR‑tree consists of four node types: S (series), P (parallel), Q (single edge), and R (rigid). Each node represents a component of C with a limited set of embedding choices.

The algorithm proceeds in several stages:

1. SPQR‑tree construction – The common graph C is decomposed into its SPQR‑tree in linear time.
2. Independent planar embeddings – For each input graph G₁ and G₂, a standard planar embedding is computed (e.g., using Hopcroft‑Tarjan). At this point the embeddings ignore the requirement that the common part be identical.
3. Extraction of embedding options – For every SPQR node the algorithm enumerates the admissible permutations of incident edges (for P‑nodes) or the fixed cyclic order of faces (for R‑nodes).
4. Constraint propagation with PQ‑trees – The requirement that shared vertices and edges have the same representation translates into ordering constraints on the permutations of each SPQR node. These constraints are encoded in a PQ‑tree, which compactly represents all permissible orderings. When a conflict arises, the PQ‑tree becomes empty, signalling that simultaneous planarity is impossible.
5. Consistency check and final embedding – If the PQ‑tree remains non‑empty after processing all nodes, a concrete choice of embedding for each SPQR component can be extracted. Combining these choices yields a simultaneous planar drawing of G₁ and G₂ that respects the common subgraph C.

Because each SPQR node and each PQ‑tree operation is visited at most once, the total running time stays linear. The authors prove correctness by showing two directions: (i) any simultaneous planar embedding induces a set of choices that satisfy all PQ‑tree constraints, and (ii) any set of choices that satisfies the constraints can be assembled into a crossing‑free simultaneous drawing. The proof also handles the subtle interaction between series/parallel components and the rigid R‑components, where the cyclic order of faces must be identical in both graphs.

Beyond the two‑graph case, the paper extends the technique to k planar graphs under a restricted overlap model: each vertex or edge is either present in all k graphs or belongs to exactly one graph. Under this model the common subgraph remains 2‑connected, so the same SPQR‑tree can be reused. The algorithm now maintains a separate PQ‑tree for each graph’s embedding options but enforces a single global set of ordering constraints. The linear‑time bound persists because the total size of all PQ‑trees is linear in the combined input size.

Experimental evaluation on synthetic random graphs and real‑world networks (e.g., power‑grid and road‑network datasets) demonstrates dramatic speedups compared with previous exponential‑search approaches. The new method solves instances with thousands of vertices in milliseconds, while earlier techniques either time out or require prohibitive memory.

In conclusion, the paper delivers the first linear‑time algorithm for simultaneous planarity when the shared graph is 2‑connected, and it shows that the approach scales to multiple graphs under a natural commonality restriction. This breakthrough narrows the gap toward a general polynomial‑time solution for simultaneous planarity and opens avenues for future work on higher connectivity, more flexible overlap models, and practical applications in multilayer network visualization and VLSI design.