Related papers: Distance-balanced graphs and Travelling Salesman P…
We prove concentration inequalities for general functions of weakly dependent random variables satisfying the Dobrushin condition. In particular, we show Talagrand's convex distance inequality for this type of dependence. We apply our…
There have lately been several suggestions for parametrized distances on a graph that generalize the shortest path distance and the commute time or resistance distance. The need for developing such distances has risen from the observation…
Consider a stationary Poisson process $\eta$ in the $d$-dimensional Euclidean or hyperbolic space and construct a random graph with vertex set $\eta$ as follows. First, each point $x\in\eta$ is connected by an edge to its nearest neighbour,…
Graph comparison deals with identifying similarities and dissimilarities between graphs. A major obstacle is the unknown alignment of graphs, as well as the lack of accurate and inexpensive comparison metrics. In this work we introduce the…
We consider the Travelling Salesman Problem with Neighbourhoods (TSPN) on the Euclidean plane ($\mathbb{R}^2$) and present a Polynomial-Time Approximation Scheme (PTAS) when the neighbourhoods are parallel line segments with lengths between…
The $c$-packedness property, proposed in 2010, is a geometric property that captures the spatial distribution of a set of edges. Despite the recent interest in $c$-packedness, its utility has so far been limited to Fr\'echet distance…
Consider a graph on $n$ uniform random points in the unit square, each pair being connected by an edge with probability $p$ if the inter-point distance is at most $r$. We show that as $n\to\infty$ the probability of full connectivity is…
A graph labeling assigns values to the components of a graph (vertices, edges, etc.). In particular, distance magic labelings have been widely studied in undirected graphs. In such a labeling, the vertices are labeled with unique values…
We aim to connect two problems, namely, dispersion and load balancing. Both problems have already been studied over static as well as dynamic graphs. Though dispersion and load balancing share some common features, the tools used in solving…
We study the Travelling Salesperson (TSP) and the Steiner Tree problem (STP) in graphs of low highway dimension. This graph parameter was introduced by Abraham et al. [SODA 2010] as a model for transportation networks, on which TSP and STP…
We propose the use of non-parametric, graph-based tests to assess the distributional balance of covariates in observational studies with multi-valued treatments. Our tests utilize graph structures ranging from Hamiltonian paths that connect…
Random geometric graphs result from taking $n$ uniformly distributed points in the unit cube, $[0,1]^d$, and connecting two points if their Euclidean distance is at most $r$, for some prescribed $r$. We show that monotone properties for…
Persistent homology (PH) has been widely applied to graph data to extract topological features. However, little attention has been paid to how different distance functions on a graph affect the resulting persistence barcodes and their…
In the companion paper [Linear rank-width of distance-hereditary graphs I. A polynomial-time algorithm, Algorithmica 78(1):342--377, 2017], we presented a characterization of the linear rank-width of distance-hereditary graphs, from which…
We show that the traveling salesman problem (TSP) and its many variants may be modeled as functional optimization problems over a graph. In this formulation, all vertices and arcs of the graph are functionals; i.e., a mapping from a space…
This paper is concerned with the linkedness of Cartesian products of complete graphs. A graph with at least $2k$ vertices is {\it $k$-linked} if, for every set of $2k$ distinct vertices organised in arbitrary $k$ pairs of vertices, there…
Consider a finite directed graph without cycles in which the arrows are weighted. We present an algorithm for the computation of a new distance, called path-length-weighted distance, which has proven useful for graph analysis in the context…
The monography considers the problem of constructing a Hamiltonian cycle in a complete graph. A rule for constructing a Hamiltonian cycle based on isometric cycles of a graph is established. An algorithm for constructing a Hamiltonian cycle…
A $2$-distance-transitive graph is a vertex-transitive graph whose vertex stabilizer is transitive on both the first step and the second step neighborhoods. In this paper, we first answer a question of A. Devillers, M. Giudici, C. H. Li and…
We consider the traveling salesperson problem in a directed graph. The pyramidal tours with step-backs are a special class of Hamiltonian cycles for which the traveling salesperson problem is solved by dynamic programming in polynomial…