Related papers: Determining the Number of Holes of a 2D Digital Co…
Based on previous results of digital topology, this paper focuses on algorithms of topological invariants of objects in 2D and 3D Digital Spaces. We specifically interest in solving hole counting of 2D objects and genus of closed surface in…
A hole in a graph is an induced subgraph which is a cycle of length at least four. We prove that for every positive integer k, every triangle-free graph with sufficiently large chromatic number contains holes of k consecutive lengths.
In two dimensional digital geometry, two lattice points are 4-connected (resp. 8-connected) if their Euclidean distance is at most one (resp. $\sqrt{2}$). A set $S \subset Z^2$ is 4-connected (resp. 8-connected) if for all pair of points…
We compute the integral homology and cohomology groups of configuration spaces of two distinct points on a given real projective space. The explicit answer is related to the (known multiplicative structure in the) integral cohomology---with…
In Digital Geometry, gaps are some basic portion of a digital object that a discrete ray can cross without intersecting any voxel of the object itself. Such a notion is quite important in combinatorial image analysis and it is strictly…
Features such as photon rings, jets, or hot. spots can leave particular topological signatures in a black hole image. As such, topological data analysis can be used to characterize images resulting from high resolution observations…
Cohomology and cohomology ring of three-dimensional (3D) objects are topological invariants that characterize holes and their relations. Cohomology ring has been traditionally computed on simplicial complexes. Nevertheless, cubical…
Let G be a simply connected absolutely simple algebraic group defined over the field of real numbers R. Let H be a simply connected semisimple R-subgroup of G. We consider the homogeneous space X=G/H. We ask: How many connected components…
Black holes are real astrophysical objects, but their interiors are hidden and can only be "observed" through mathematics. The structure of rotating black holes is typically illustrated with the help of special coordinates. But any such…
Given a set of points $S \subseteq \mathbb{R}^2$, a subset $X \subseteq S$ with $|X|=k$ is called $k$-gon if all points of $X$ lie on the boundary of the convex hull of $X$, and $k$-hole if, in addition, no point of $S \setminus X$ lies in…
We compute the integral cohomology ring of configuration spaces of two points on a given real projective space. Apart from an integral class, the resulting ring is a quotient of the known integral cohomology of the dihedral group of order 8…
Digital topological methods are often used on computing the topological complexity of digital images. We give new results on the relation between reducibility and digital contractibility in order to determine the topological complexity of a…
A cube is an 8-rep-tile: it is the union of eight smaller copies of itself. Is there a set with a hole which has this property? The computer found an interesting and complicated solution, which then could be simplified. We discuss some…
A hole of a simple connected graph $G$ is a chordless cycle $C_n,$ where $n \in \Bbb N, n \geq 4,$ in the graph $G$. The girth of a simple connected graph $G$ is the smallest cycle in $G$, if any such cycle exists. It can be observed that…
A two-dimensional configuration is a coloring of the infinite grid Z^2 with finitely many colors. For a finite subset D of Z^2, the D-patterns of a configuration are the colored patterns of shape D that appear in the configuration. The…
Let D be an acyclic digraph. The competition graph of D is a graph which has the same vertex set as D and has an edge between x and y if and only if there exists a vertex v in D such that (x,v) and (y,v) are arcs of D. For any graph G, G…
We give a complete solution to the extremal topological combinatorial problem of finding the minimum number of tiles needed to construct a polyomino with $h$ holes. We denote this number by $g(h)$ and say that a polyomino is crystallized if…
A hole is an induced cycle of length at least 4, and an odd hole is a hole of odd length. It is NP-hard to color the vertices of an odd hole-free graph. A graph $G$ is perfectly divisible if every induced subgraph $H$ of $G$ with at least…
As a generalization of the linking number, we construct a set of invariant numbers for two-component handlebody-links. These numbers are elementary divisors associated with the natural homomorphism from the first homology group of a…
Call a pure Hodge structure geometric if it is contained in the cohomology of a smooth complex projective variety. The main goal is to show that for any set of Hodge numbers (subject to the obvious constraints), there exists a geometric…