Testing Graph Isotopy on Surfaces

Abstract

We investigate the following problem: Given two embeddings G 1 and G 2 of the same abstract graph G on an orientable surface S, decide whether G 1 and G 2 are isotopic; in other words, whether there exists a continuous family of embeddings between G 1 and G 2 .We provide efficient algorithms to solve this problem in two models. In the first model, the input consists of the arrangement of G 1 (resp., G 2 ) with a fixed graph cellularly embedded on S; our algorithm is linear in the input complexity, and thus, optimal. In the second model, G 1 and G 2 are piecewise-linear embeddings in the plane minus a finite set of points; our algorithm runs in O(n 3/2 log n) time, where n is the complexity of the input.The graph isotopy problem is a natural variation of the homotopy problem for closed curves on surfaces and on the punctured plane, for which algorithms have been given by various authors; we use some of these algorithms as a subroutine.As a by-product, we reprove the following mathematical characterization, first observed by Ladegaillerie (1984): Two graph embeddings are isotopic if and only if they are homotopic and congruent by an oriented homeomorphism. * This article is the full version of a conference article bearing the same title, which appeared in the Proceedings of the Twenty-Eighth Symposium on Computational Geometry, 2012. We acknowledge partial support from the French ANR Blanc project ANR-12-BS02-005 (RDAM).

Mitochondrial networks through the lens of mathematics

“…To remedy this, we can use the topological concept of isotopy with fixed vertices, which (loosely) is a continuous deformation of one graph to another that preserves the relative connectivity of the edges and vertices while not introducing new loops or edge crossings [133]. Determining whether two embedded graphs are isotopic can likely be accomplished with a linear-time algorithm based on established work [108,[134][135][136]. Related and likely informative approaches also include homotopy groups, persistent homotopy, and analysis of polynomial invariants of mitochondrial-like networks [137][138][139][140][141][142][143].…”

Planar and Toroidal Morphs Made Easier

“…Following Chambers et al [12], we represent geodesic torus drawings using a coordinate representation (P, τ) that that records Two crossing-free drawings of the same graph on the torus are isotopic if one can be deformed into the other through a continuous family of (not necessarily geodesic) crossing-free drawings. Two crossing-free drawings are isotopic if and only if their coordinate representations can be normalized so that their translation vectors agree; this condition can be tested in O(n) time [12, Theorem A.1], [15]. A geodesic isotopy or morph is an isotopy in which all intermediate drawings are geodesic and crossing-free.…”

Embedding Half-Edge Graphs in Punctured Surfaces

“…Cellular embeddings of graphs in punctured surfaces -We address now the embeddings in a punctured surface of a graph. Note that the following definition of a cellular embedding of graphs in punctured surfaces differs from that of [8]. We ensure, for instance, that the embedding occurs in open surfaces while, in that work, the boundary is included in the topological space of the surface.…”

“…The number of parts of this splitting is precisely the number of half-edges of its corresponding external cycle. Finally, a half-edge uniquely corresponds to a HR and, in each external cycle of G H , the number of open faces is equal to the number of HR, see (8).…”

“…Graphs embedded in surfaces have been studied in different contexts with applications ranging from combinatorics, geometry to computer science (see [12] and the reviews [8] and [9] for a detailed account on this active subject). The correspondence between embeddings of graphs in surfaces and ribbon graphs can be traced back to the work by Heffter in [16].…”