Recognizing Weak Embeddings of Graphs
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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 ψ_ε : G→ M with φ-ψ_ε<ε for every ε>0. A polynomial-time algorithm for recognizing weak embeddings was recently found by [Fulek and Kynčl, 2017], which reduces to solving a system of linear equations over Z_2. It runs in O(n^2ω)≤ O(n^4.75) 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 n). Our algorithm is also conceptually simpler than [Fulek and Kynčl, 2017]: 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 [Akitaya et al., 2016], and combines local constraints on the orientation of subgraphs directly, thereby eliminating the need for solving large systems of linear equations.
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