Recognizing Weak Embeddings of Graphs

We present an efficient algorithm for a problem in the interface between clustering and graph embeddings. An embedding φ : G→M of a graph G into a 2-manifold M maps the vertices in V (G) to distinct points and the edges in E(G) to interior-disjoint Jordan arcs between the corresponding vertices. In applications in clustering, cartography, and visualization, nearby vertices and edges are often bundled to a common node or arc, due to data compression or low resolution. This raises the computational problem of deciding whether a given map φ : G→M comes from an embedding. A map φ : G → M is a weak embedding if it can be perturbed into an embedding ψε : G→M with ‖φ− ψε‖ < ε for every ε > 0. A polynomial-time algorithm for recognizing weak embeddings was recently found by Fulek and Kynčl [14], which reduces to solving a system of linear equations over Z2. It runs in O(n) ≤ O(n) time, where ω ≈ 2.373 is the matrix multiplication exponent and n is the number of vertices and edges of G. We improve the running time to O(n log n). Our algorithm is also conceptually simpler than [14]: We perform a sequence of local operations that gradually “untangles” the image φ(G) into an embedding ψ(G), or reports that φ is not a weak embedding. It generalizes a recent technique developed for the case that G is a cycle and the embedding is a simple polygon [1], and combines local constraints on the orientation of subgraphs directly, thereby eliminating the need for solving large systems of linear equations.