Hanani-Tutte for Approximating Maps of Graphs

We resolve in the affirmative conjectures of A. Skopenkov and Repovs (1998), and M. 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

Unified Hanani Tutte theorem

We introduce a common generalization of the strong Hanani–Tutte theorem and the weak Hanani–Tutte theorem: if a graph G has a drawing D in the plane where every pair of independent edges crosses an even number of times, then G has a planar drawing preserving the rotation of each vertex whose incident edges cross each other evenly in D. The theorem is implicit in the proof of the strong Hanani–Tutte theorem by Pelsmajer, Schaefer and Štefankovič. We give a new, somewhat simpler proof

Unified Hanani Tutte theorem

We introduce a common generalization of the strong Hanani–Tutte theorem and the weak Hanani–Tutte theorem: if a graph G has a drawing D in the plane where every pair of independent edges crosses an even number of times, then G has a planar drawing preserving the rotation of each vertex whose incident edges cross each other evenly in D. The theorem is implicit in the proof of the strong Hanani–Tutte theorem by Pelsmajer, Schaefer and Štefankovič. We give a new, somewhat simpler proof

Embedding Graphs into Embedded Graphs

A (possibly degenerate) drawing of a graph G in the plane is approximable by an embedding if it can be turned into an embedding by an arbitrarily small perturbation. We show that testing, whether a drawing of a planar graph G in the plane is approximable by an embedding, can be carried out in polynomial time, if a desired embedding of G belongs to a fixed isotopy class, i.e., the rotation system (or equivalently the faces) of the embedding of G and the choice of outer face are fixed. In other words, we show that c-planarity with embedded pipes is tractable for graphs with fixed embeddings. To the best of our knowledge an analogous result was previously known essentially only when G is a cycle

Strong Hanani-Tutte for the Torus

If a graph can be drawn on the torus so that every two independent edges cross an even number of times, then the graph can be embedded on the torus

LIPIcs

The genus g(G) of a graph G is the minimum g such that G has an embedding on the orientable surface M_g of genus g. A drawing of a graph on a surface is independently even if every pair of nonadjacent edges in the drawing crosses an even number of times. The Z_2-genus of a graph G, denoted by g_0(G), is the minimum g such that G has an independently even drawing on M_g. By a result of Battle, Harary, Kodama and Youngs from 1962, the graph genus is additive over 2-connected blocks. In 2013, Schaefer and Stefankovic proved that the Z_2-genus of a graph is additive over 2-connected blocks as well, and asked whether this result can be extended to so-called 2-amalgamations, as an analogue of results by Decker, Glover, Huneke, and Stahl for the genus. We give the following partial answer. If G=G_1 cup G_2, G_1 and G_2 intersect in two vertices u and v, and G-u-v has k connected components (among which we count the edge uv if present), then |g_0(G)-(g_0(G_1)+g_0(G_2))|= m >= 3, we prove that g_0(K_{m,n})/g(K_{m,n})=1-O(1/n). Similar results are proved also for the Euler Z_2-genus. We express the Z_2-genus of a graph using the minimum rank of partial symmetric matrices over Z_2; a problem that might be of independent interest

C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width

For a clustered graph, i.e, a graph whose vertex set is recursively partitioned into clusters, the C-Planarity Testing problem asks whether it is possible to find a planar embedding of the graph and a representation of each cluster as a region homeomorphic to a closed disk such that 1. the subgraph induced by each cluster is drawn in the interior of the corresponding disk, 2. each edge intersects any disk at most once, and 3. the nesting between clusters is reflected by the representation, i.e., child clusters are properly contained in their parent cluster. The computational complexity of this problem, whose study has been central to the theory of graph visualization since its introduction in 1995 [Qing-Wen Feng, Robert F. Cohen, and Peter Eades. Planarity for clustered graphs. ESA'95], has only been recently settled [Radoslav Fulek and Csaba D. T\'oth. Atomic Embeddability, Clustered Planarity, and Thickenability. To appear at SODA'20]. Before such a breakthrough, the complexity question was still unsolved even when the graph has a prescribed planar embedding, i.e, for embedded clustered graphs. We show that the C-Planarity Testing problem admits a single-exponential single-parameter FPT algorithm for embedded clustered graphs, when parameterized by the carving-width of the dual graph of the input. This is the first FPT algorithm for this long-standing open problem with respect to a single notable graph-width parameter. Moreover, in the general case, the polynomial dependency of our FPT algorithm is smaller than the one of the algorithm by Fulek and T\'oth. To further strengthen the relevance of this result, we show that the C-Planarity Testing problem retains its computational complexity when parameterized by several other graph-width parameters, which may potentially lead to faster algorithms.Comment: Extended version of the paper "C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width" to appear in the Proceedings of the 14th International Symposium on Parameterized and Exact Computation (IPEC 2019

Lifting generic maps to embeddings

Given a generic PL map or a generic smooth fold map $f:N^n\to M^m$, where $m\ge n$ and $2(m+k)\ge 3(n+1)$, we prove that $f$ lifts to a PL or smooth embedding $N\to M\times\mathbb R^k$ if and only if its double point locus $(f\times f)^{-1}(\Delta_M)\setminus\Delta_N$ admits an equivariant map to $S^{k-1}$. As a corollary we answer a 1990 question of P. Petersen on whether the universal coverings of the lens spaces $L(p,q)$, $p$ odd, lift to embeddings in $L(p,q)\times\mathbb R^3$. We also show that if a non-degenerate PL map $N\to M$ lifts to a topological embedding in $M\times\mathbb R^k$ then it lifts to a PL embedding in there. The Appendix extends the 2-multi-0-jet transversality over the usual compactification of $M\times M\setminus\Delta_M$ and Section 3 contains an elementary theory of stable PL maps.Comment: 37 pages. v4: Added a discussion of stable PL maps (in Section 3) and the general case of the extended 2-multi-0-jet transversality theorem (in the end of the Appendix