33 research outputs found
Tree-width of hypergraphs and surface duality
In Graph Minors III, Robertson and Seymour write: "It seems that the
tree-width of a planar graph and the tree-width of its geometric dual are
approximately equal - indeed, we have convinced ourselves that they differ by
at most one". They never gave a proof of this. In this paper, we prove a
generalisation of this statement to embedding of hypergraphs on general
surfaces, and we prove that our bound is tight
Dynamic programming for graphs on surfaces
We provide a framework for the design and analysis of dynamic
programming algorithms for surface-embedded graphs on n vertices
and branchwidth at most k. Our technique applies to general families
of problems where standard dynamic programming runs in 2O(k·log k).
Our approach combines tools from topological graph theory and
analytic combinatorics.Postprint (updated version
The -genus of Kuratowski minors
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
-genus of a graph is the minimum such that has an
independently even drawing on the orientable surface of genus . An
unpublished result by Robertson and Seymour implies that for every , every
graph of sufficiently large genus contains as a minor a projective
grid or one of the following so-called -Kuratowski graphs: , or
copies of or sharing at most common vertices. We show that
the -genus of graphs in these families is unbounded in ; in
fact, equal to their genus. Together, this implies that the genus of a graph is
bounded from above by a function of its -genus, solving a problem
posed by Schaefer and \v{S}tefankovi\v{c}, and giving an approximate version of
the Hanani-Tutte theorem on orientable surfaces. We also obtain an analogous
result for Euler genus and Euler -genus of graphs.Comment: 23 pages, 7 figures; a few references added and correcte
On Hardness of the Joint Crossing Number
The Joint Crossing Number problem asks for a simultaneous embedding of two
disjoint graphs into one surface such that the number of edge crossings
(between the two graphs) is minimized. It was introduced by Negami in 2001 in
connection with diagonal flips in triangulations of surfaces, and subsequently
investigated in a general form for small-genus surfaces. We prove that all of
the commonly considered variants of this problem are NP-hard already in the
orientable surface of genus 6, by a reduction from a special variant of the
anchored crossing number problem of Cabello and Mohar
Dynamic Programming for Graphs on Surfaces
We provide a framework for the design and analysis of dynamic programming
algorithms for surface-embedded graphs on n vertices and branchwidth at most k.
Our technique applies to general families of problems where standard dynamic
programming runs in 2^{O(k log k)} n steps. Our approach combines tools from
topological graph theory and analytic combinatorics. In particular, we
introduce a new type of branch decomposition called "surface cut
decomposition", generalizing sphere cut decompositions of planar graphs
introduced by Seymour and Thomas, which has nice combinatorial properties.
Namely, the number of partial solutions that can be arranged on a surface cut
decomposition can be upper-bounded by the number of non-crossing partitions on
surfaces with boundary. It follows that partial solutions can be represented by
a single-exponential (in the branchwidth k) number of configurations. This
proves that, when applied on surface cut decompositions, dynamic programming
runs in 2^{O(k)} n steps. That way, we considerably extend the class of
problems that can be solved in running times with a single-exponential
dependence on branchwidth and unify/improve most previous results in this
direction.Comment: 28 pages, 3 figure
Shortest path embeddings of graphs on surfaces
The classical theorem of F\'{a}ry states that every planar graph can be
represented by an embedding in which every edge is represented by a straight
line segment. We consider generalizations of F\'{a}ry's theorem to surfaces
equipped with Riemannian metrics. In this setting, we require that every edge
is drawn as a shortest path between its two endpoints and we call an embedding
with this property a shortest path embedding. The main question addressed in
this paper is whether given a closed surface S, there exists a Riemannian
metric for which every topologically embeddable graph admits a shortest path
embedding. This question is also motivated by various problems regarding
crossing numbers on surfaces.
We observe that the round metrics on the sphere and the projective plane have
this property. We provide flat metrics on the torus and the Klein bottle which
also have this property.
Then we show that for the unit square flat metric on the Klein bottle there
exists a graph without shortest path embeddings. We show, moreover, that for
large g, there exist graphs G embeddable into the orientable surface of genus
g, such that with large probability a random hyperbolic metric does not admit a
shortest path embedding of G, where the probability measure is proportional to
the Weil-Petersson volume on moduli space.
Finally, we construct a hyperbolic metric on every orientable surface S of
genus g, such that every graph embeddable into S can be embedded so that every
edge is a concatenation of at most O(g) shortest paths.Comment: 22 pages, 11 figures: Version 3 is updated after comments of
reviewer