Hanani-Tutte for approximating maps of graphs

Abstract

We resolve in the affirmative conjectures of Repovš andA. Skopenkov (1998), andM. Skopenkov (2003) generalizing the classical Hanani-Tutte theorem to the setting of approximating maps of graphs on 2-dimensional surfaces by embeddings. Our proof of this result is constructive and almost immediately implies an efficient algorithm for testing whether a given piecewise linear map of a graph in a surface is approximable by an embedding. More precisely, an instance of this problem consists of (i) a graph G whose vertices are partitioned into clusters and whose inter-cluster edges are partitioned into bundles, and (ii) a region R of a 2-dimensional compact surface M given as the union of a set of pairwise disjoint discs corresponding to the clusters and a set of pairwise disjoint "pipes" corresponding to the bundles, connecting certain pairs of these discs. We are to decide whether G can be embedded inside M so that the vertices in every cluster are drawn in the corresponding disc, the edges in every bundle pass only through its corresponding pipe, and every edge crosses the boundary of each disc at most once.

Recognizing Weak Embeddings of Graphs

“…In the plane (i.e., M = R 2 ), only planar graphs admit weak embeddings hence m = O(n), but our techniques work for 2-manifolds of arbitrary genus, and G may be a dense graph. Our result improves the running time of the previous algorithm [13] from O(m 2ω ) ≤ O(m 4.75 ) to O(m log m), where ω ∈ [2, 2.373) is the matrix multiplication exponent. It also improves the running times of several recent polynomialtime algorithms in special cases, e.g., when the embedding of G is restricted to a given isotopy class [12], and H is a path [2] (see below).…”

“…Finding efficient algorithms for the recognition of weak embeddings ϕ : G → H, where G is an arbitrary graph, was posed as an open problem in [1,5,6]. The first polynomial-time solution for the general version follows from a recent variant [13] of the Hanani-Tutte theorem [14,23], which was conjectured by M. Skopenkov [21] in 2003 and in a slightly weaker form already by Repovš and A. Skopenkov [18] in 1998. However, this algorithm reduces the problem to a system of O(m) linear equations over Z 2 , where m = |E(G)|.…”

“…Outline. Our results rely on ideas from [2,5,6] and [13]. To distinguish the graphs G and H, we use the convention that G has vertices and edges, and H has clusters and pipes.…”

“…We may further assume that ψ ε maps every edge uv to a Jordan arc that crosses the boundaries of D u (resp., D v ) precisely once (cf. [5,Lemma B2] and [13,Section 4]).…”

Recognizing Weakly Simple Polygons

“…The Fréchet distance between two drawings, P and Q, of H is defined as dist F (P, Q) = inf φ:H→H max x∈H dist(P (φ(x)), Q(x)), where φ is an automorphism of H (a homeomorphism from H to itself). Very recently, Fulek and Kynčl [13] gave a polynomial-time algorithm for deciding whether a given drawing of a graph H is weakly simple, i.e., whether a straight-line drawing P of H is within ε Fréchet distance from some embedding Q of H, for all ε > 0. Earlier, efficient algorithms were known only in special cases: when the embedding is restricted to a given isotopy class (i.e., given combinatorial embedding) [12]; and when all n vertices are collinear and the isotopy class is given [1].…”