Topology Preserving Edge Contraction Paper By Dr Tamal

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 dtuple of real numbers. Norm of a Point – If the point x = (x 1, x 2, x 3…xd), the norm ||x|| = (Sxi 2)1/2 Euclidean Space – A d-dimensional Euclidean space Rd is the set of ddimensional points together with the euclidean distance function mapping each set of points (x, y) to ||x-y||.

More Definitions d– 1 sphere: Sd-1 = {x Î Rd | ||x|| = 1} 1 -Sphere – Circle, 2 -Sphere (hollow) d-ball: Bd = {x Î Rd | ||x|| £ 1} 2 -ball - Disk (curve+interior), 3 -ball – Sphere (Solid) The surface of a d-ball is a d-1 sphere. d-halfspace: Hd = {x Î Rd | x 1 = 1}

Even More Definitions Manifold: A d-manifold is a nonempty topological space where at each point, the neighborhood is either a Rd or a Hd. With Boundary/ Without Boundary

Lots more Definitions k-Simplex is the convex hull of k+1 affinely independent point k ³ 0

Still more Definitions Face: If s is a simplex a face of s, t is defined by a non-empty subset of the p 0 k+1 points. p 3 p 1 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} p 2 Proper faces

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

Simplicial Complex A collection of simplices, K, such that if s Î K and t £ s, then t Î K i. e. for each face in K, all the faces of it is there K and all their subfaces are there etc. and s, s’ Î K => l sÇs’ = f or l sÇs’ £ s and sÇs’ £ s’ i. e. if two faces intersect, they intersect on their face.

Simplicial Complex p 0 p 3 p 1 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} Examples of a non-simplicial complex: {p 0, p 0 p 1} p 2 p 0 Examples of a non-simplicial complex: p 4 {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} p 3 p 1 p 2

Subcomplex, Closure A subcomplex of a simplicial complex one of its subsets that is a simplicial complex in itself. {p 0, p 1, p 0 p 1} 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| = UsÎK int s

Closure Let B Í 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. p 0 p 1 p 2

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

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, L

Subdivision A subdivision of K is a complex Sd K such that |Sd K| = |K| and s Î K => s Î Sd K

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

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 f: |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 s, a simplex in K, if dim St s = k, the order is the smallest interger I for which there is a (k-i) simplex h such that St s ~ St h What is that mumbo-jumbo? ?

Order (contd. )

Boundary The Jth boundary of a simplicial complex K is the set of simplices with order no less than j. Order Bound: Jth boundary can contain only simplices of dimensions not more than dim K-j Jth 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 jab: |K| |L| defined by a surjective u if u Î Vert K – {a, b} vertexf(u) map = c if u Î {a, b} Outside |St ab|, the mapping is unity. Inside, it is not even injective.

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