Related papers: Reconsideration of Tangle and Ultrafilter using Se…
Traditional clustering identifies groups of objects that share certain qualities. Tangles do the converse: they identify groups of qualities that often occur together. They can thereby discover, relate, and structure types: of behaviour,…
Graphene, dubbed as a two-dimensional material represents the topological concept of "surface" embedded in a three-dimensional space. This regard enables to employ existing theories/tools in topology to understand different…
Fractional (hyper-)graph theory is concerned with the specific problems that arise when fractional analogues of otherwise integer-valued (hyper-)graph invariants are considered. The focus of this paper is on fractional edge covers of…
In finite graphs, finite-order tangles offer an abstract description of highly connected substructures. In infinite graphs, infinite-order tangles compactify the graphs in the same way the ends compactify connected locally finite graphs.…
Traditional clustering identifies groups of objects that share certain qualities. Tangles do the converse: they identify groups of qualities that often occur together. They can thereby identify and discover 'types': of behaviour, views,…
A thrackle is a drawing of a graph in which each pair of edges meets precisely once. Conway's Thrackle Conjecture asserts that a thrackle drawing of a graph on the plane cannot have more edges than vertices. We prove the Conjecture for…
A tangle of order $k$ in a matroid or graph may be thought of as a "$k$-connected component". For a tangle of order $k$ in a matroid or graph that satisfies a certain robustness condition, we describe a tree decomposition of the matroid or…
This paper defines, for each graph $G$, a flag vector $fG$. The flag vectors of the graphs on $n$ vertices span a space whose dimension is $p(n)$, the number of partitions on $n$. The analogy with convex polytopes indicates that the linear…
In this paper we study the bilateral filter proposed by Tomasi and Manduchi, as a spectral domain transform defined on a weighted graph. The nodes of this graph represent the pixels in the image and a graph signal defined on the nodes…
Superpixel segmentation consists of partitioning images into regions composed of similar and connected pixels. Its methods have been widely used in many computer vision applications since it allows for reducing the workload, removing…
Robertson and Seymour proved two fundamental theorems about tangles in graphs: the tree-of-tangles theorem, which says that every graph has a tree-decomposition such that distinguishable tangles live in different nodes of the tree, and the…
The ability to compare complex systems can provide new insight into the fundamental nature of the processes captured in ways that are otherwise inaccessible to observation. Here, we introduce the $n$-tangle method to directly compare two…
We survey an abstract theory of connectivity, based on symmetric submodular set functions. We start by developing Robertson and Seymour's fundamental duality between branch decompositions (related to the better-known tree decompositions)…
We identify and study a simple combinatorial problem that is derived from submodularity issues encountered in the theory of tangles of graphs and abstract separation systems.
We study tangle replacement in the context of spatial graphs. The main results show that, for certain spatial handcuff graphs, there is a one-to-one correspondence between the neighborhood equivalence classes of the spatial graphs obtained…
In image processing, a segmentation is a process of partitioning an image into multiple sets of pixels, that are defined as super-pixels. Each super-pixel is characterized by a label or parameter. Here, we are proposing a method for…
A divisibility relation on ultrafilters is defined as follows: ${\cal F}\hspace{1mm}\widetilde{\mid}\hspace{1mm}{\cal G}$ if and only if every set in $\cal F$ upward closed for divisibility also belongs to $\cal G$. After describing the…
Classical path search assumes complete graphs and scalar optimization metrics, yet real infrastructure networks are incomplete and require multi-dimensional evaluation. We introduce the concept of traversal: a generalization of paths that…
Linear-width is a well-known and highly regarded graph parameter. The concept of Single Ideal and Linear obstacle serves as an obstruction to linear-width on a connectivity sysem. In this concise paper, we present an alternative proof for…
In the last decade it became apparent that a large number of the most interesting structures and phenomena of the world can be described by networks: separable elements, with connections (or interactions) between certain pairs of them.…