scholarly journals Shattering, Graph Orientations, and Connectivity

10.37236/3326 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
László Kozma ◽  
Shay Moran
Keyword(s):  
Graph Theory ◽  
Combinatorial Optimization ◽  
Main Tool ◽  
The Other ◽  
Vc Dimension ◽  
Forbidden Subgraphs ◽  
Set Systems ◽  
Flows In Networks ◽  
Extremal Systems

We present a connection between two seemingly disparate fields: VC-theory and graph theory. This connection yields natural correspondences between fundamental concepts in VC-theory, such as shattering and VC-dimension, and well-studied concepts of graph theory related to connectivity, combinatorial optimization, forbidden subgraphs, and others.In one direction, we use this connection to derive results in graph theory. Our main tool is a generalization of the Sauer-Shelah Lemma (Pajor, 1985; Bollobás and Radcliffe, 1995; Dress, 1997; Holzman and Aharoni). Using this tool we obtain a series of inequalities and equalities related to properties of orientations of a graph. Some of these results appear to be new, for others we give new and simple proofs.In the other direction, we present new illustrative examples of shattering-extremal systems - a class of set-systems in VC-theory whose understanding is considered by some authors to be incomplete Bollobás and Radcliffe, 1995; Greco, 1998; Rónyai and Mészáros, 2011). These examples are derived from properties of orientations related to distances and flows in networks.

scholarly journals The VC Dimension of Metric Balls under Fréchet and Hausdorff Distances

2021 ◽  
Author(s):  
Anne Driemel ◽  
André Nusser ◽  
Jeff M. Phillips ◽  
Ioannis Psarros
Keyword(s):  
Density Estimation ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Similarity Metrics ◽  
Vc Dimension ◽  
Set Systems ◽  
Polygonal Curves ◽  
The Individual ◽  
Curve Similarity ◽  
Metric Balls

AbstractThe Vapnik–Chervonenkis dimension provides a notion of complexity for systems of sets. If the VC dimension is small, then knowing this can drastically simplify fundamental computational tasks such as classification, range counting, and density estimation through the use of sampling bounds. We analyze set systems where the ground set X is a set of polygonal curves in $$\mathbb {R}^d$$ R d and the sets $$\mathcal {R}$$ R are metric balls defined by curve similarity metrics, such as the Fréchet distance and the Hausdorff distance, as well as their discrete counterparts. We derive upper and lower bounds on the VC dimension that imply useful sampling bounds in the setting that the number of curves is large, but the complexity of the individual curves is small. Our upper and lower bounds are either near-quadratic or near-linear in the complexity of the curves that define the ranges and they are logarithmic in the complexity of the curves that define the ground set.

Framed 4-valent graph minor theory I: Introduction. A planarity criterion and linkless embeddability

2014 ◽  
Vol 23 (07) ◽  
pp. 1460002
Author(s):  
Vassily Olegovich Manturov
Keyword(s):  
Graph Theory ◽  
The Other ◽  
Graph Minor ◽  
Other Hand ◽  
Simple Class ◽  
Points Of View ◽  
General Graphs ◽  
Planarity Criterion

This paper is the first one in the sequence of papers about a simple class of framed 4-graphs; the goal of this paper is to collect some well-known results on planarity and to reformulate them in the language of minors. The goal of the whole sequence is to prove analogs of the Robertson–Seymour–Thomas theorems for framed 4-graphs: namely, we shall prove that many minor-closed properties are classified by finitely many excluded graphs. From many points of view, framed 4-graphs are easier to consider than general graphs; on the other hand, framed 4-graphs are closely related to many problems in graph theory.

The ontology of person-centered healthcare: Theoretical essentials to reground medicine within its humanistic framework. Part I - A centered concept of personhood: Some tools to conceptualise subjectivity

2018 ◽  
Vol 6 (1) ◽  
pp. 146
Author(s):  
Thomas Frölich ◽  
F F Bevier ◽  
Alicja Babakhani ◽  
Hannah H Chisholm ◽  
Peter Henningsen ◽  
...  
Keyword(s):  
Graph Theory ◽  
Free Space ◽  
Rooted Tree ◽  
The Other ◽  
Discrete Mathematics ◽  
Soap Bubble ◽  
Soap Bubbles ◽  
In Series ◽  
The One ◽  
Individual Trees

To address subjectivity, as a generally rooted phenomenon, other ways of visualisation must be applied than in conventional objectivistic approaches. Using ‘trees’ as operational metaphors, as employed in Arthur Cayley’s ‘theory of the analytical forms called trees’, one rooted ‘tree’ must be set beneath the other and, if such ‘trees’ are combined, the resulting ‘forest’ is nevertheless made up of individual ‘trees’ and not of a deconstructed mix of ‘roots’, ‘branches’, ‘leaves’ or further categories, each understood as addressable both jointly and individually. The reasons for why we have chosen a graph theory and corresponding discrete mathematics as an approach and application are set out in this first of our three articles. It combines two approaches that, in combination, are quite uncommon and which are therefore not immediately familiar to all readers. But as simple as it is to imagine a tree, or a forest, it is equally simple to imagine a child blowing soap bubbles with the aid of a blow ring. A little more challenging, perhaps, is the additional idea of arranging such blow rings in series, transforming the size of the soap bubble in one ring after the other. To finally combine both pictures, the one of trees and the other of blow rings, goes beyond simple imagination, especially when we prolong the imagined blow ring becoming a tunnel, with a specific inner shape. The inner shape of the blow ring and its expansion as a tunnel are understood as determined by discrete qualities, each forming an internal continuity, depicted as a scale, with the scales combined in the form of a glyph plot. The different positions on these scales determine their length and if the endpoints of the spines are connected with an enveloping line then this corresponds to the free space left open in the tunnel to go through it. Using so many visualisation techniques at once is testing. Nevertheless, this is what we propose here and to facilitate such a visualisation within the imagination, we do it step by step. As the intended result of this ‘juggling of three balls’, as it were, we end up with a concept of how living beings elaborate their principal structure to enable controlled outside-inside communication.

scholarly journals Clustering Multiple Hydrographs Using Mathematical Optimization

10.29007/6r61 ◽  
2018 ◽  
Author(s):  
Kazuhiro Matsumoto ◽  
Mamoru Miyamoto
Keyword(s):  
Combinatorial Optimization ◽  
Water Level ◽  
Continuous Optimization ◽  
Mathematical Optimization ◽  
Optimization Procedure ◽  
The Other ◽  
Flood Events ◽  
Number Of Clusters ◽  
Different Characteristics ◽  
Insight Into

A mathematical optimization procedure is presented to group multiple hydrographs into a small number of clusters for the purpose of helping to understand various runoff behaviors observed in flood events in a basin. In grouping, the hydrographs belonging to each cluster can be estimated within the specified accuracy by the corresponding parameter set. The effectiveness is demonstrated using twenty-seven hydrographs observed in nine flood events and at three water level stations in the Abe River basin in Japan. The optimization results illustrate that eight sets of parameters are necessary to estimate such hydrographs within the specified accuracy. One parameter set commonly estimates as many as seven out of twenty-seven hydrographs while some other parameter sets estimate the other hydrographs with different characteristics specific to flood events or water level stations. Most of the previous research is based on continuous optimization; however, a presenting procedure such as clustering is based on combinatorial optimization. Thus, new insight into understanding the runoff behaviors is brought by combinatorial optimization which is not often used in previous research.

scholarly journals Un modello basato sulla teoria dei grafi per l’individuazione di isomorfismi distrettuali geograficamente non prossimi. Il caso dei distretti industriali italiani

2021 ◽  
pp. 27-50
Author(s):  
Stefano De Falco
Keyword(s):  
Graph Theory ◽  
The Other ◽  
Industrial Districts ◽  
Industrial District ◽  
Other Hand ◽  
The One ◽  
Practical Implications

A dichotomy often frequent in the context of geographic studies concerns the dualism between propagation and induction models-based phenomena, inherent in variables and factors characterizing contiguous areas, and research relating to homogeneity between geographically not closed areas. In the wake of the latter research, this contribution proposes a model that exploits the potential of graph theory for the evaluation of common dynamics relating to non-contiguous areas. The assumption underlying the model envisages configuring the reality being studied in terms of a network whose nodes and branches are respectively representative of entities distant from each other and of their related affinities. The proposed approach focuses on some Italian industrial districts. The value of the proposed approach is twofold, on the one hand regarding the specific industrial district topic with both scientific and practical implications, and on the other hand it aims to provide a method that can be replicated in similar scenarios in which it is interesting to evaluate the similarity between neighboring areas analytically.

scholarly journals Forcing $k$-Repetitions in Degree Sequences

10.37236/3503 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Yair Caro ◽  
Asaf Shapira ◽  
Raphael Yuster
Keyword(s):  
Graph Theory ◽  
Positive Integer ◽  
Main Tool ◽  
Geometric Approach ◽  
Degree Sequences ◽  
Integer Sequences ◽  
Sum Theorem ◽  
Zero Sum

One of the most basic results in graph theory states that every graph with at least two vertices has two vertices with the same degree. Since there are graphs without $3$ vertices of the same degree, it is natural to ask if for any fixed $k$, every graph $G$ is "close" to a graph $G'$ with  $k$ vertices of the same degree. Our main result in this paper is that this is indeed the case. Specifically, we show that for any positive integer $k$, there is a constant $C=C(k)$, so that given any graph $G$, one can remove from $G$ at most $C$ vertices and thus obtain a new graph $G'$ that contains at least $\min\{k,|G|-C\}$ vertices of the same degree.Our main tool is a multidimensional zero-sum theorem for integer sequences, which we prove using an old geometric approach of Alon and Berman.

scholarly journals A Note on Path Factors of $(3,4)$-Biregular Bipartite Graphs

10.37236/705 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Carl Johan Casselgren
Keyword(s):  
Graph Theory ◽  
Bipartite Graph ◽  
Edge Coloring ◽  
Bipartite Graphs ◽  
The Other ◽  
Spanning Subgraph ◽  
Interval Coloring ◽  
Proper Edge Coloring

A proper edge coloring of a graph $G$ with colors $1,2,3,\dots$ is called an interval coloring if the colors on the edges incident with any vertex are consecutive. A bipartite graph is $(3,4)$-biregular if all vertices in one part have degree $3$ and all vertices in the other part have degree $4$. Recently it was proved [J. Graph Theory 61 (2009), 88-97] that if such a graph $G$ has a spanning subgraph whose components are paths with endpoints at 3-valent vertices and lengths in $\{2, 4, 6, 8\}$, then $G$ has an interval coloring. It was also conjectured that every simple $(3,4)$-biregular bipartite graph has such a subgraph. We provide some evidence for this conjecture by proving that a simple $(3,4)$-biregular bipartite graph has a spanning subgraph whose components are nontrivial paths with endpoints at $3$-valent vertices and lengths not exceeding $22$.

scholarly journals HOW MUCH DOES ANTICIPATION COST TO CHARLES BAUDELAIRE?

2020 ◽  
pp. 59-64
Author(s):  
M. S. Filippovich ◽  
Keyword(s):  
Walter Benjamin ◽  
Main Tool ◽  
Charles Baudelaire ◽  
The Other ◽  
Other Hand

Walter Benjamin noticed an anticipation of Charles Baudelaire that gets the modern attributes. In the article, the anticipation is analyzed as a part of the existential presence of a poet. By this way the poet can resist the fate and he has a feeling of future. Baudelaire’s literature career and his lifestyle are an embodiment of it. It is the main tool to stand with a faith. From one hand, his works have no similarity with others poets, in their loneliness, flânerie, and manners, from the other hand in their personal embodiment.

scholarly journals A variant of Niessen’s problem on degreesequences of graphs

2014 ◽  
Vol Vol. 16 no. 1 (Graph Theory) ◽  
Author(s):  
Jiyun Guo ◽  
Jianhua Yin
Keyword(s):  
Graph Theory ◽  
Discrete Math ◽  
Simple Graph ◽  
The Other ◽  
Degree Sequences ◽  
International Audience

Graph Theory International audience Let (a1,a2,\textellipsis,an) and (b1,b2,\textellipsis,bn) be two sequences of nonnegative integers satisfying the condition that b1>=b2>=...>=bn, ai<= bi for i=1,2,\textellipsis,n and ai+bi>=ai+1+bi+1 for i=1,2,\textellipsis, n-1. In this paper, we give two different conditions, one of which is sufficient and the other one necessary, for the sequences (a1,a2,\textellipsis,an) and (b1,b2,\textellipsis,bn) such that for every (c1,c2,\textellipsis,cn) with ai<=ci<=bi for i=1,2,\textellipsis,n and &#x2211;&limits;i=1n ci=0 (mod 2), there exists a simple graph G with vertices v1,v2,\textellipsis,vn such that dG(vi)=ci for i=1,2,\textellipsis,n. This is a variant of Niessen\textquoterights problem on degree sequences of graphs (Discrete Math., 191 (1998), 247&#x2013;253).

scholarly journals Shattering-Extremal Set Systems of VC Dimension at most 2

10.37236/4548 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Tamás Mészáros ◽  
Lajos Rónyai
Keyword(s):  
Vc Dimension ◽  
Set Systems ◽  
Dimension 2 ◽  
Extremal Set Systems ◽  
Extremal Set ◽  
Open Question

We say that a set system $\mathcal{F}\subseteq 2^{[n]}$ shatters a given set $S\subseteq [n]$ if $2^S=\{F~\cap~S ~:~F~\in~\mathcal{F}\}$. The Sauer inequality states that in general, a set system $\mathcal{F}$ shatters at least $|\mathcal{F}|$ sets. Here we concentrate on the case of equality. A set system is called shattering-extremal if it shatters exactly $|\mathcal{F}|$ sets. In this paper we characterize shattering-extremal set systems of Vapnik-Chervonenkis dimension $2$ in terms of their inclusion graphs, and as a corollary we answer an open question about leaving out elements from shattering-extremal set systems in the case of families of Vapnik-Chervonenkis dimension $2$.

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