On n-fold L(j,k)-and circular L(j,k)-labelings of graphs

AbstractWe initiate research on the multiple distance 2 labeling of graphs in this paper.Let n,j,k be positive integers. An n-fold L(j,k)-labeling of a graph G is an assignment f of sets of nonnegative integers of order n to the vertices of G such that, for any two vertices u,v and any two integers a∈f(u), b∈f(v), |a−b|≥j if uv∈E(G), and |a−b|≥k if u and v are distance 2 apart. The span of f is the absolute difference between the maximum and minimum integers used by f. The n-fold L(j,k)-labeling number of G is the minimum span over all n-fold L(j,k)-labelings of G.Let n,j,k and m be positive integers. An n-fold circular m-L(j,k)-labeling of a graph G is an assignment f of subsets of {0,1,…,m−1} of order n to the vertices of G such that, for any two vertices u,v and any two integers a∈f(u), b∈f(v), min{|a−b|,m−|a−b|}≥j if uv∈E(G), and min{|a−b|,m−|a−b|}≥k if u and v are distance 2 apart. The minimum m such that G has an n-fold circular m-L(j,k)-labeling is called the n-fold circular L(j,k)-labeling number of G.We investigate the basic properties of n-fold L(j,k)-labelings and circular L(j,k)-labelings of graphs. The n-fold circular L(j,k)-labeling numbers of trees, and the hexagonal and p-dimensional square lattices are determined. The upper and lower bounds for the n-fold L(j,k)-labeling numbers of trees are obtained. In most cases, these bounds are attainable. In particular, when k=1 both the lower and the upper bounds are sharp. In many cases, the n-fold L(j,k)-labeling numbers of the hexagonal and p-dimensional square lattices are determined. In other cases, upper and lower bounds are provided. In particular, we obtain the exact values of the n-fold L(j,1)-labeling numbers of the hexagonal and p-dimensional square lattices

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

Introduction to computer image processing

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Novel Aggregated Solutions for Robust Visual Tracking in Traffic Scenarios

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Symmetry in Graph Theory

This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view

Combinatorial Optimization

This report summarizes the meeting on Combinatorial Optimization where new and promising developments in the field were discussed. Th

Pixel-Level Deep Multi-Dimensional Embeddings for Homogeneous Multiple Object Tracking

The goal of Multiple Object Tracking (MOT) is to locate multiple objects and keep track of their individual identities and trajectories given a sequence of (video) frames. A popular approach to MOT is tracking by detection consisting of two processing components: detection (identification of objects of interest in individual frames) and data association (connecting data from multiple frames). This work addresses the detection component by introducing a method based on semantic instance segmentation, i.e., assigning labels to all visible pixels such that they are unique among different instances. Modern tracking methods often built around Convolutional Neural Networks (CNNs) and additional, explicitly-defined post-processing steps. This work introduces two detection methods that incorporate multi-dimensional embeddings. We train deep CNNs to produce easily-clusterable embeddings for semantic instance segmentation and to enable object detection through pose estimation. The use of embeddings allows the method to identify per-pixel instance membership for both tasks. Our method specifically targets applications that require long-term tracking of homogeneous targets using a stationary camera. Furthermore, this method was developed and evaluated on a livestock tracking application which presents exceptional challenges that generalized tracking methods are not equipped to solve. This is largely because contemporary datasets for multiple object tracking lack properties that are specific to livestock environments. These include a high degree of visual similarity between targets, complex physical interactions, long-term inter-object occlusions, and a fixed-cardinality set of targets. For the reasons stated above, our method is developed and tested with the livestock application in mind and, specifically, group-housed pigs are evaluated in this work. Our method reliably detects pigs in a group housed environment based on the publicly available dataset with 99% precision and 95% using pose estimation and achieves 80% accuracy when using semantic instance segmentation at 50% IoU threshold. Results demonstrate our method\u27s ability to achieve consistent identification and tracking of group-housed livestock, even in cases where the targets are occluded and despite the fact that they lack uniquely identifying features. The pixel-level embeddings used by the proposed method are thoroughly evaluated in order to demonstrate their properties and behaviors when applied to real data. Adivser: Lance C. Pére