Topology Preserving Edge Contraction Paper By Dr. Tamal Dey et al Presented by Ramakrishnan Kazhiyur-Mannar

Some Definitions (Lots actually) Point – a d-dimensional point is a d-tuple of real numbers. Norm of a Point – If the point x = (x 1, x 2, x 3 …x d ), the norm ||x|| = (  x i 2 ) 1/2 Euclidean Space – A d-dimensional Euclidean space R d is the set of d-dimensional points together with the euclidean distance function mapping each set of points (x,y) to ||x-y||.

More Definitions d–1 sphere: S d-1 = {x  R d | ||x|| = 1} 1-Sphere – Circle, 2-Sphere-Sphere (hollow) d-ball: B d = {x  R d | ||x||  1} 2-ball - Disk (curve+interior), 3-ball – Sphere (Solid) The surface of a d-ball is a d-1 sphere. d-halfspace: H d = {x  R d | x 1 = 1}

Even More Definitions Manifold:A d-manifold is a non-empty topological space where at each point, the neighborhood is either a R d or a H d. With Boundary/ Without Boundary

Lots more Definitions k-Simplex is the convex hull of k+1 affinely independent point k  0

Still more Definitions Face: If  is a simplex a face of ,  is defined by a non-empty subset of the k+1 points. Proper faces Example of faces: {p 0 }, {p 1 }, {p 0, p 0 p 1 }, {p 0, p 1, p 2, p 0 p 1, p 0 p 2, p 1 p 2, p 0 p 1 p 2 } p0p0 p1p1 p2p2 p3p3

Definitions (I have given up trying to get unique titles) Coface: If  is a face of , then  is a coface of , written as . The interior of the simplex is the set of points contained in  but not on any proper face of .

Simplicial Complex A collection of simplices, K, such that if  K and , then  K i.e. for each face in K, all the faces of it is there K and all their subfaces are there etc. and   ’  K =>  ’  or  ’  and  ’  ’ i.e. if two faces intersect, they intersect on their face.

Examples of a non-simplicial complex: {p 0, p 1, p 2, p 3, p 4, p 0 p 1, p 1 p 2, p 2 p 0, p 3 p 4 } Simplicial Complex Examples of a simplicial complex: {p 0 }, {p 0, p 1, p 2, p 0 p 1 } {p 0, p 1, p 2, p 0 p 1, p 0 p 2, p 1 p 2, p 0 p 1 p 2 } p0p0 p1p1 p2p2 p3p3 Examples of a non-simplicial complex: {p 0, p 0 p 1 } p0p0 p1p1 p2p2 p4p4 p3p3

Subcomplex, Closure A subcomplex of a simplicial complex one of its subsets that is a simplicial complex in itself. {p0, p1, p0p1} is a subcomplex of {p 0, p 1, p 2, p 0 p 1, p 1 p 2, p 2 p 0, p 0 p 1 p 2 } The Underlying space is the union of simplex interiors. |K| = U  K int 

Closure Let B  K (B need not be a subcomplex). Closure of B is the set of all faces of simplices of B. The Closure is the smallest subcomplex that contains B. p0p0 p1p1 p2p2

Star The star of B is the set of all cofaces of simplices in B.

Link Link of B is the set of all faces of cofaces of simplices in B that are disjoint from the simples in B

Mathematically Speaking Or Simply, 

Subdivision A subdivision of K is a complex Sd K such that |Sd K| = |K| and  K =>  Sd K

Homeomorphism Homeomorphism is topological equivalence An intuitive definition? Technical definition: Homeomorphism between two spaces X and Y is a bijection h: X  Y such that both h and h’ are continuous. If  a Homeomorphism between two spaces then they are homeomorphic X  Y and are said to be of the same topological type or genus.

Combinatorial Version Complexes stand for topological spaces in combinatorial domain. A vertex map for two complexes K and L is a function f: Vert K  Vert L. A Simplicial Map  : |K|  |L| is defined by

Combinatorial Version (contd.) f need not be injective or surjective. It is a homeomorphism iff f is bijective and f -1 is a vertex map. Here, we call it isomorphism denoted by K  L. There is a slight difference between isomorphism and homeomorphism.

Order Remember manifolds? What if the neighborhood of a point is not a ball? For  a simplex in K, if dim St  = k, the order is the smallest interger I for which there is a (k-i) simplex  such that St s  St h What is that mumbo-jumbo??

Order (contd.)

Boundary The J th boundary of a simplicial complex K is the set of simplices with order no less than j. Order Bound: J th boundary can contain only simplices of dimensions not more than dim K-j J th boundary contains (j+1) st Boundary. This is used to have a hierarchy of complexes.

Edge Contraction (Finally!!)

In the Language of Math… Contraction is a surjective simplicial map  ab :|K|  |L| defined by a surjective vertex map Outside |St ab|, the mapping is unity. Inside, it is not even injective. f(u) = u if u  Vert K – {a, b} c if u  {a, b}

One Last Term… An unfolding  of  ab is a simplicial homeomorphism |K|  |L|. It is local if it differs from  ab only inside |E| and it is relaxed if it differs from  ab only inside |St E| Now, WHAT IS THAT??!!!

How do I get there? Basically, the underlying space should not be affected in order to maintain topology.

So, What IS the Condition?! Simple. If I were to overlay the two stars, the links must be the same! The condition is: Lk a  Lk b = Lk ab

Finally, THANKS!!! Wake up now!!