31,304 research outputs found
Towards an Isomorphism Dichotomy for Hereditary Graph Classes
In this paper we resolve the complexity of the isomorphism problem on all but
finitely many of the graph classes characterized by two forbidden induced
subgraphs. To this end we develop new techniques applicable for the structural
and algorithmic analysis of graphs. First, we develop a methodology to show
isomorphism completeness of the isomorphism problem on graph classes by
providing a general framework unifying various reduction techniques. Second, we
generalize the concept of the modular decomposition to colored graphs, allowing
for non-standard decompositions. We show that, given a suitable decomposition
functor, the graph isomorphism problem reduces to checking isomorphism of
colored prime graphs. Third, we extend the techniques of bounded color valence
and hypergraph isomorphism on hypergraphs of bounded color size as follows. We
say a colored graph has generalized color valence at most k if, after removing
all vertices in color classes of size at most k, for each color class C every
vertex has at most k neighbors in C or at most k non-neighbors in C. We show
that isomorphism of graphs of bounded generalized color valence can be solved
in polynomial time.Comment: 37 pages, 4 figure
Link groups of 4-manifolds
The notion of a Bing cell is introduced, and it is used to define invariants,
link groups, of 4-manifolds. Bing cells combine some features of both surfaces
and 4-dimensional handlebodies, and the link group \lambda(M) measures certain
aspects of the handle structure of a 4-manifold M. This group is a quotient of
the fundamental group, and examples of manifolds are given with \pi_1(M) not
equal to \lambda(M). The main construction of the paper is a generalization of
the Milnor group, which is used to formulate an obstruction to embeddability of
Bing cells into 4-space. Applications to the A-B slice problem and to the
structure of topological arbiters are discussed.Comment: 34 pages, 7 figures. v.3: minor phrasing change
Phase transitions in 3D gravity and fractal dimension
We show that for three dimensional gravity with higher genus boundary
conditions, if the theory possesses a sufficiently light scalar, there is a
second order phase transition where the scalar field condenses. This three
dimensional version of the holographic superconducting phase transition occurs
even though the pure gravity solutions are locally AdS. This is in addition
to the first order Hawking-Page-like phase transitions between different
locally AdS handlebodies. This implies that the R\'enyi entropies of
holographic CFTs will undergo phase transitions as the R\'enyi parameter is
varied, as long as the theory possesses a scalar operator which is lighter than
a certain critical dimension. We show that this critical dimension has an
elegant mathematical interpretation as the Hausdorff dimension of the limit set
of a quotient group of AdS, and use this to compute it, analytically near
the boundary of moduli space and numerically in the interior of moduli space.
We compare this to a CFT computation generalizing recent work of Belin, Keller
and Zadeh, bounding the critical dimension using higher genus conformal blocks,
and find a surprisingly good match
Largeness and SQ-universality of cyclically presented groups
Largeness, SQ-universality, and the existence of free subgroups of rank 2 are measures of the complexity of a finitely presented group. We obtain conditions under which a cyclically presented group possesses one or more of these properties. We apply our results to a class of groups introduced by Prishchepov which contains, amongst others, the various generalizations of Fibonacci groups introduced by Campbell and Robertson
- …