4,243 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
A spectral lower bound for the divisorial gonality of metric graphs
Let be a compact metric graph, and denote by the Laplace
operator on with the first non-trivial eigenvalue . We
prove the following Yang-Li-Yau type inequality on divisorial gonality
of . There is a universal constant such that
where the
volume is the total length of the edges in ,
is the minimum length of all the geodesic paths
between points of of valence different from two, and is the
largest valence of points of . Along the way, we also establish
discrete versions of the above inequality concerning finite simple graph models
of and their spectral gaps.Comment: 22 pages, added new recent references, minor revisio
A Fixed Parameter Tractable Approximation Scheme for the Optimal Cut Graph of a Surface
Given a graph cellularly embedded on a surface of genus , a
cut graph is a subgraph of such that cutting along yields a
topological disk. We provide a fixed parameter tractable approximation scheme
for the problem of computing the shortest cut graph, that is, for any
, we show how to compute a approximation of
the shortest cut graph in time .
Our techniques first rely on the computation of a spanner for the problem
using the technique of brick decompositions, to reduce the problem to the case
of bounded tree-width. Then, to solve the bounded tree-width case, we introduce
a variant of the surface-cut decomposition of Ru\'e, Sau and Thilikos, which
may be of independent interest
A Turaev surface approach to Khovanov homology
We introduce Khovanov homology for ribbon graphs and show that the Khovanov
homology of a certain ribbon graph embedded on the Turaev surface of a link is
isomorphic to the Khovanov homology of the link (after a grading shift). We
also present a spanning quasi-tree model for the Khovanov homology of a ribbon
graph.Comment: 30 pages, 18 figures, added sections on virtual links and
Reidemeister move
From rubber bands to rational maps: A research report
This research report outlines work, partially joint with Jeremy Kahn and
Kevin Pilgrim, which gives parallel theories of elastic graphs and conformal
surfaces with boundary. One one hand, this lets us tell when one rubber band
network is looser than another, and on the other hand tell when one conformal
surface embeds in another.
We apply this to give a new characterization of hyperbolic critically finite
rational maps among branched self-coverings of the sphere, by a positive
criterion: a branched covering is equivalent to a hyperbolic rational map if
and only if there is an elastic graph with a particular "self-embedding"
property. This complements the earlier negative criterion of W. Thurston.Comment: 52 pages, numerous figures. v2: New example
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