Sphere Graph Research Articles

Spotted disk and sphere graphs. II

The disk graph of a handlebody H of genus g ≥ 2 with m ≥ 0 marked points on the boundary is the graph whose vertices are isotopy classes of disks disjoint from the marked points and where two vertices are connected by an edge of length one if they can be realized disjointly. We show that for m = 2 the disk graph contains quasi-isometrically embedded copies of ℝ2. Furthermore, the sphere graph of the doubled handlebody of genus g ≥ 4 with two marked points contains for every n ≥ 1 a quasi-isometrically embedded copy of ℝn.

Projections of the sphere graph to the arc graph of a surface

Let [Formula: see text] be a compact surface, and [Formula: see text] be the double of a handlebody. Given a homotopy class of maps from [Formula: see text] to [Formula: see text] inducing an isomorphism of fundamental groups, we describe a canonical uniformly Lipschitz retraction of the sphere graph of [Formula: see text] to the arc graph of [Formula: see text]. We also show that this retraction is a uniformly bounded distance from the nearest point projection map.

Uniform fellow traveling between surgery paths in the sphere graph

We show that the Hausdorff distance between any forward and any backward surgery paths in the sphere graph is at most 2. From this it follows that the Hausdorff distance between any two surgery paths with the same initial sphere system and same target sphere system is at most 4. Our proof relies on understanding how surgeries affect the Guirardel core associated to sphere systems. We show that applying a surgery is equivalent to performing a Rips move on the Guirardel core.

View Sphere Partitioning via Flux Graphs Boosts Recognition from Sparse Views

View-based 3D object recognition requires a selection of model object views against which to match a query view. Ideally, for this to be computationally efficient, such a selection should be sparse. To address this problem we partition the view sphere into regions within which the silhouette of a model object is qualitatively unchanged. This is accomplished using a flux-based skeletal representation and skeletal matching to compute the pairwise similarity between two views. Associating each view with a node of a view sphere graph, with the similarity between a pair of views as an edge weight, a clustering algorithm is used to partition the view sphere. Our experiments on exemplar level recognition using 19 models from the Toronto Database and category level recognition using 150 models from the McGill Shape Benchmark demonstrate that in a scenario of recognition from sparse views, sampling model views from such partitions consistently boosts recognition performance when compared against queries sampled randomly or uniformly from the view sphere. We demonstrate the improvement in recognition accuracy for a variety of popular 2D shape similarity approaches: shock graph matching, flux graph matching, shape context based matching and inner distance based matching.

Realisation and dismantlability

We study dismantling properties of the arc, disc and sphere graphs. We prove that any finite subgroup H of the mapping class group of a surface with punctures, the handlebody group, or Out(F_n) fixes a filling (resp. simple) clique in the appropriate graph. We deduce realisation theorems, in particular the Nielsen Realisation Problem in the case of a nonempty set of punctures. We also prove that infinite H have either empty or contractible fixed point sets in the corresponding complexes. Furthermore, we show that their spines are classifying spaces for proper actions for mapping class groups and Out(F_n).

Graph manifolds, left-orderability and amalgamation

We show that every irreducible toroidal integer homology sphere graph manifold has a left-orderable fundamental group. This is established by way of a specialization of a result due to Bludov and Glass for the almagamated products that arise, and in this setting work of Boyer, Rolfsen and Wiest may be applied. Our result then depends on input from 3-manifold topology and Heegaard Floer homology.

Splice diagram determining singularity links and universal abelian covers

To a rational homology sphere graph manifold one can associate a weighted tree invariant called splice diagram. In this article we prove a sufficient numerical condition on the splice diagram for a graph manifold to be a singularity link. We also show that if two manifolds have the same splice diagram, then their universal abelian covers are homeomorphic. To prove the last theorem we have to generalize our notions to orbifolds.

The [formula omitted]-labeling of [formula omitted]-free graphs and its applications

An L ( 2 , 1 ) -labeling of a graph G is a function f from the vertex set V ( G ) into the set of nonnegative integers such that | f ( x ) − f ( y ) | ≥ 2 if d ( x , y ) = 1 and | f ( x ) − f ( y ) | ≥ 1 if d ( x , y ) = 2 , where d ( x , y ) denotes the distance between x and y in G . The L ( 2 , 1 ) -labeling number, λ ( G ) , of G is the minimum k where G has an L ( 2 , 1 ) -labeling f with k being the absolute difference between the largest and smallest image points of f . In this work, we will study the L ( 2 , 1 ) -labeling on K 1 , n -free graphs where n ≥ 3 and apply the result to unit sphere graphs which are of particular interest in the channel assignment problem.

Classification of Stokes Graphs of Second Order Fuchsian Differential Equations of Genus Two

Stokes curves of second order Fuchsian dfferential equations on the Riemann sphere with a large parameter form sphere graphs, which are called Stokes graphs. Topological classification of Stokes graphs are given for the case where equations have five regular singular points. It is proved that there are exactly 25 degree sequences of sphere triangulations associated with Stokes graphs under suitable generic conditions.

Unit disk graph recognition is NP-hard

Unit disk graphs are the intersection graphs of unit diameter closed disks in the plane. This paper gives a polynomial-time reduction from SATISFIABILITY to the problem of recognizing unit disk graphs. Equivalently, it shows that determining if a graph has sphericity 2 or less, even if the graph is planar or is known to have sphericity at most 3, is NP-hard. We show how this reduction can be extended to 3 dimensions, thereby showing that unit sphere graph recognition, or determining if a graph has sphericity 3 or less, is also NP-hard. We conjecture that K-sphericity is NP-hard for all fixed K greater than 1.