423 research outputs found
Surface Split Decompositions and Subgraph Isomorphism in Graphs on Surfaces
The Subgraph Isomorphism problem asks, given a host graph G on n vertices and
a pattern graph P on k vertices, whether G contains a subgraph isomorphic to P.
The restriction of this problem to planar graphs has often been considered.
After a sequence of improvements, the current best algorithm for planar graphs
is a linear time algorithm by Dorn (STACS '10), with complexity .
We generalize this result, by giving an algorithm of the same complexity for
graphs that can be embedded in surfaces of bounded genus. At the same time, we
simplify the algorithm and analysis. The key to these improvements is the
introduction of surface split decompositions for bounded genus graphs, which
generalize sphere cut decompositions for planar graphs. We extend the algorithm
for the problem of counting and generating all subgraphs isomorphic to P, even
for the case where P is disconnected. This answers an open question by Eppstein
(SODA '95 / JGAA '99)
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
JSJ decompositions of Quadratic Baumslag-Solitar groups
Generalized Baumslag-Solitar groups are defined as fundamental groups of
graphs of groups with infinite cyclic vertex and edge groups. Forester proved
(in "On uniqueness of JSJ decompositions of finitely generated groups",
Comment. Math. Helv. 78 (2003) pp 740-751) that in most cases the defining
graphs are cyclic JSJ decompositions, in the sense of Rips and Sela. Here we
extend Forester's results to graphs of groups with vertex groups that can be
either infinite cyclic or quadratically hanging surface groups.Comment: 20 pages, 2 figures. Several corrections and improvements from
referee's report. Imprtant changes in Definition 5.1, and the proof of
Theorem 5.5 (previously 5.4). Lemma 5.4 was adde
Profinite complexes of curves, their automorphisms and anabelian properties of moduli stacks of curves
Let , for , be the D-M moduli stack of smooth
curves of genus labeled by unordered distinct points. The main result
of the paper is that a finite, connected \'etale cover {\cal M}^\l of , defined over a sub--adic field , is "almost" anabelian in
the sense conjectured by Grothendieck for curves and their moduli spaces.
The precise result is the following. Let \pi_1({\cal M}^\l_{\ol{k}}) be the
geometric algebraic fundamental group of {\cal M}^\l and let
{Out}^*(\pi_1({\cal M}^\l_{\ol{k}})) be the group of its exterior
automorphisms which preserve the conjugacy classes of elements corresponding to
simple loops around the Deligne-Mumford boundary of {\cal M}^\l (this is the
"-condition" motivating the "almost" above). Let us denote by
{Out}^*_{G_k}(\pi_1({\cal M}^\l_{\ol{k}})) the subgroup consisting of
elements which commute with the natural action of the absolute Galois group
of . Let us assume, moreover, that the generic point of the D-M stack
{\cal M}^\l has a trivial automorphisms group. Then, there is a natural
isomorphism: {Aut}_k({\cal M}^\l)\cong{Out}^*_{G_k}(\pi_1({\cal
M}^\l_{\ol{k}})). This partially extends to moduli spaces of curves the
anabelian properties proved by Mochizuki for hyperbolic curves over
sub--adic fields.Comment: This paper has been withdrawn because of a flaw in the paper
"Profinite Teichm\"uller theory" of the first author, on which this paper
built o
Finite rigid sets in sphere complexes
A subcomplex of a simplicial complex is rigid if every
locally injective, simplicial map is the restriction of an
automorphism of . Aramayona and the second author proved that the
curve complex of an orientable surface can be exhausted by finite rigid sets.
The Hatcher sphere complex is an analog of the curve complex for isotopy
classes of essential spheres in a connect sum of copies of .
We show that there is an exhaustion of the sphere complex by finite rigid sets
for all and that when the sphere complex does not have finite
rigid sets.Comment: 13 pages, 5 figure
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