Computer Science

This tutorial review provides a guiding reference to researchers who want to have an overview of the large body of literature about graph spanners. It reviews the current literature covering various research streams about graph spanners, such as different formulations, sparsity and lightness results, computational complexity, dynamic algorithms, and applications. As an additional contribution, we offer a list of open problems on graph spanners.

Risk classification plays an important role in many regulations and standards. However, a general method that provides an optimal classification has not been proposed yet. Also, the criteria of optimality are not defined in these regulations. In this work, we will propose a mathematical model that is sufficient to describe this problem, and we also propose an algorithm that classifies graph vertices based on their risk value in polynomial time.

Novel Monte Carlo estimators are proposed to solve both the Tikhonov regularization (TR) and the interpolation problems on graphs. These estimators are based on random spanning forests (RSF), the theoretical properties of which enable to analyze the estimators' theoretical mean and variance. We also show how to perform hyperparameter tuning for these RSF-based estimators. TR is a component in many well-known algorithms, and we show how the proposed estimators can be easily adapted to avoid expensive intermediate steps in generalized semi-supervised learning, label propagation, Newton's method and iteratively reweighted least squares. In the experiments, we illustrate the proposed methods on several problems and provide observations on their run time.

An equitable coloring of a graph G=(V,E) is a (proper) vertex-coloring of G , such that the sizes of any two color classes differ by at most one. In this paper, we consider the equitable coloring problem in block graphs. Recall that the latter are graphs in which each 2-connected component is a complete graph. The problem remains hard in the class of block graphs. In this paper, we present some graph theoretic results relating various parameters. Then we use them in order to trace some algorithmic implications, mainly dealing with the fixed-parameter tractability of the problem.

In this paper, we introduce a new method which we call it MZ-method, for dividing a natural number x by two and then we use graph as a model to show MZ-algorithm. Applying (recursively) k -times of the MZ-method for the number x , produces a graph with unique structure that is denoted by G k (x) . We investigate the structure of G k (x) . Also from the natural number x and graph G k (x) we produce codes which are important and applicable in the information security and cryptography.

A set of n points in the plane which are not all collinear defines at least n distinct lines. Chen and Chvátal conjectured in 2008 that a similar result can be achieved in the broader context of finite metric spaces. This conjecture remains open even for graph metrics. In this article we prove that graphs with no induced house nor induced cycle of length at least~5 verify the desired property. We focus on lines generated by vertices at distance at most 2, define a new notion of ``good pairs'' that might have application in larger families, and finally use a discharging technique to count lines in irreducible graphs.

Wiener index, defined as the sum of distances between all unordered pairs of vertices, is one of the most popular molecular descriptors. It is well known that among 2-vertex connected graphs on n≥3 vertices, the cycle C n attains the maximum value of Wiener index. We show that the second maximum graph is obtained from C n by introducing a new edge that connects two vertices at distance two on the cycle if n≠6 . If n≥11 , the third maximum graph is obtained from a 4 -cycle by connecting opposite vertices by a path of length n−3 . We completely describe also the situation for n≤10 .

A proper labelling of a graph G is a pair (π, c π ) in which π is an assignment of numeric labels to some elements of G , and c π is a colouring induced by π through some mathematical function over the set of labelled elements. In this work, we consider gap-vertex-labellings, in which the colour of a vertex is determined by a function considering the largest difference between the labels assigned to its neighbours. We present the first upper-bound for the vertex-gap number of arbitrary graphs, which is the least number of labels required to properly label a graph. We investigate families of graphs which do not admit any gap-vertex-labelling, regardless of the number of labels. Furthermore, we introduce a novel parameter associated with this labelling and provide bounds for it for complete graphs K n .

An n -length binary word is q -decreasing, q≥1 , if every of its length maximal factor of the form 0 a 1 b satisfies a=0 or q⋅a>b .We show constructively that these words are in bijection with binary words having no occurrences of 1 q+1 , and thus they are enumerated by the (q+1) -generalized Fibonacci numbers. We give some enumerative results and reveal similarities between q -decreasing words and binary words having no occurrences of 1 q+1 in terms of frequency of 1 bit. In the second part of our paper, we provide an efficient exhaustive generating algorithm for q -decreasing words in lexicographic order, for any q≥1 , show the existence of 3-Gray codes and explain how a generating algorithm for these Gray codes can be obtained. Moreover, we give the construction of a more restrictive 1-Gray code for 1 -decreasing words, which in particular settles a conjecture stated recently in the context of interconnection networks by Eğecioğlu and Iršič.

In the group-testing literature, efficient algorithms have been developed to minimize the number of tests required to identify all minimal "defective" sub-groups embedded within a larger group, using deterministic group splitting with a generalized binary search. In a separate literature, researchers have used a stochastic group splitting approach to efficiently sample from the intractable number of minimal defective sets of outages in electrical power systems that trigger large cascading failures, a problem in which positive tests can be much more computationally costly than negative tests. In this work, we generate test problems with variable numbers of defective sets and a tunable positive:negative test cost ratio to compare the efficiency of deterministic and stochastic adaptive group splitting algorithms for identifying defective edges in hypergraphs. For both algorithms, we show that the optimal initial group size is a function of both the prevalence of defective sets and the positive:negative test cost ratio. We find that deterministic splitting requires fewer total tests but stochastic splitting requires fewer positive tests, such that the relative efficiency of these two approaches depends on the positive:negative test cost ratio. We discuss some real-world applications where each of these algorithms is expected to outperform the other.