Safety in s-t Paths, Trails and Walks

Given a directed graph G and a pair of nodes s and t, an s-tbridge of G is an edge whose removal breaks all s-t paths of G (and thus appears in all s-t paths). Computing all s-t bridges of G is a basic graph problem, solvable in linear time. In this paper, we consider a natural generalisation of this problem, with the notion of “safety” from bioinformatics. We say that a walk W is safe with respect to a set $${\mathcal {W}}$$ W of s-t walks, if W is a subwalk of all walks in $${\mathcal {W}}$$ W . We start by considering the maximal safe walks when $${\mathcal {W}}$$ W consists of: all s-t paths, all s-t trails, or all s-t walks of G. We show that the solutions for the first two problems immediately follow from finding all s-t bridges after incorporating simple characterisations. However, solving the third problem requires non-trivial techniques for incorporating its characterisation. In particular, we show that there exists a compact representation computable in linear time, that allows outputting all maximal safe walks in time linear in their length. Our solutions also directly extend to multigraphs, except for the second problem, which requires a more involved approach. We further generalise these problems, by assuming that safety is defined only with respect to a subset of visible edges. Here we prove a dichotomy between the s-t paths and s-t trails cases, and the s-t walks case: the former two are NP-hard, while the latter is solvable with the same complexity as when all edges are visible. We also show that the same complexity results hold for the analogous generalisations of s-tarticulation points (nodes appearing in all s-t paths). We thus obtain the best possible results for natural “safety”-generalisations of these two fundamental graph problems. Moreover, our algorithms are simple and do not employ any complex data structures, making them ideal for use in practice.

On equitable chromatic topological indices of some Mycielski graphs

In recent years, the notion of chromatic Zagreb indices has been introduced and studied for certain basic graph classes, as a coloring parallel of different Zagreb indices. A proper coloring C of a graph G, which assigns colors to the vertices of G such that the numbers of vertices in any two color classes differ by at most one, is called an equitable coloring of G. In this paper, we introduce the equitable chromatic Zagreb indices and equitable chromatic irregularity indices of some special classes of graphs called Mycielski graphs of paths and cycles.

Knowledge Graph Question Answering Using Graph-Pattern Isomorphism

Knowledge Graph Question Answering (KGQA) systems are often based on machine learning algorithms, requiring thousands of question-answer pairs as training examples or natural language processing pipelines that need module fine-tuning. In this paper, we present a novel QA approach, dubbed TeBaQA. Our approach learns to answer questions based on graph isomorphisms from basic graph patterns of SPARQL queries. Learning basic graph patterns is efficient due to the small number of possible patterns. This novel paradigm reduces the amount of training data necessary to achieve state-of-the-art performance. TeBaQA also speeds up the domain adaption process by transforming the QA system development task into a much smaller and easier data compilation task. In our evaluation, TeBaQA achieves state-of-the-art performance on QALD-8 and delivers comparable results on QALD-9 and LC-QuAD v1. Additionally, we performed a fine-grained evaluation on complex queries that deal with aggregation and superlative questions as well as an ablation study, highlighting future research challenges.

On Bi-gram Graph Attributes

We propose a new approach to text semantic analysis and general corpus analysis using, as termed in this article, a "bi-gram graph" representation of a corpus. The different attributes derived from graph theory are measured and analyzed as unique insights or against other corpus graphs, attributes such as the graph chromatic number and the graph coloring, graph density and graph K-core. We observe a vast domain of tools and algorithms that can be developed on top of the graph representation; creating such a graph proves to be computationally cheap, and much of the heavy lifting is achieved via basic graph calculations. Furthermore, we showcase the different use-cases for the bi-gram graphs and how scalable it proves to be when dealing with large datasets.

Multiset-based assessment of vulnerability of energy infrastructures to destructive impacts

This paper is dedicated to the application of the multigrammatical framework to the assessment of vulnerability of energy infrastructures affected by impacts destroying (reducing capabilities of) their facilities (power plants, fuel producing plants, power transmission lines, fuel transporting pipes, as well as networking devices of both electricity and fuel subsystems of an energy infrastructures). A basic graph representation of energy infrastructures is considered, and technique of their multigrammatical representation is introduced. Criterial base for recognition of the energy infrastructures vulnerability, being a generalization of the similar criterial base developed regarding industrial infrastructures is proposed. Techniques of multigrammatical modelling reservation of energy infrastructures and their recovery after impacts is proposed. Directions of future research in this area are announced.

Application of Basic Graph Theory in Autonomous Motion of Robots

Discrete mathematics covers the field of graph theory, which solves various problems in graphs using algorithms, such as coloring graphs. Part of graph theory is focused on algorithms that solve the passage through mazes and labyrinths. This paper presents a study conducted as part of a university course focused on graph theory. The course addressed the problem of high student failure in the mazes and labyrinths chapter. Students’ theoretical knowledge and practical skills in solving algorithms in the maze were low. Therefore, the use of educational robots and their involvement in the teaching of subjects in part focused on mazes and labyrinths. This study shows an easy passage through the individual areas of teaching the science, technology, engineering, and mathematics (STEM) concept. In this article, we describe the research survey and focus on the description and examples of teaching in a university course. Part of the work is the introduction of an easy transition from the theoretical solution of algorithms to their practical implementation on a real autonomous robot. The theoretical part of the course introduced the issues of graph theory and basic algorithms for solving the passage through the labyrinth. The contribution of this study is a change in the approach to teaching graph theory and a greater interconnection of individual areas of STEM to achieve better learning outcomes for science students.

Mixed covering arrays on graphs of small treewidth

Covering arrays are combinatorial objects that have been successfully applied in design of test suites for testing systems such as software, hardware, and networks where failures can be caused by the interaction between their parameters. Let [Formula: see text] and [Formula: see text] be positive integers with [Formula: see text]. Two vectors [Formula: see text] and [Formula: see text] are qualitatively independent if for any ordered pair [Formula: see text], there exists an index [Formula: see text] such that [Formula: see text]. Let [Formula: see text] be a graph with [Formula: see text] vertices [Formula: see text] with respective vertex weights [Formula: see text]. A mixed covering array on[Formula: see text] , denoted by [Formula: see text], is a [Formula: see text] array such that row [Formula: see text] corresponds to vertex [Formula: see text], entries in row [Formula: see text] are from [Formula: see text]; and if [Formula: see text] is an edge in [Formula: see text], then the rows [Formula: see text] are qualitatively independent. The parameter [Formula: see text] is the size of the array. Given a weighted graph [Formula: see text], a mixed covering array on [Formula: see text] with minimum size is optimal. In this paper, we introduce some basic graph operations to provide constructions for optimal mixed covering arrays on the family of graphs with treewidth at most three.

Random-Walk Laplacian for Frequency Analysis in Periodic Graphs

This paper presents the benefits of using the random-walk normalized Laplacian matrix as a graph-shift operator and defines the frequencies of a graph by the eigenvalues of this matrix. A criterion to order these frequencies is proposed based on the Euclidean distance between a graph signal and its shifted version with the transition matrix as shift operator. Further, the frequencies of a periodic graph built through the repeated concatenation of a basic graph are studied. We show that when a graph is replicated, the graph frequency domain is interpolated by an upsampling factor equal to the number of replicas of the basic graph, similarly to the effect of zero-padding in digital signal processing.

Simplex Transformations and the Multiway Cut Problem

We consider multiway cut, a basic graph partitioning problem in which the goal is to find the minimum weight collection of edges disconnecting a given set of special vertices called terminals. Multiway cut admits a well-known simplex embedding relaxation, where rounding this embedding is equivalent to partitioning the simplex. Current best-known solutions to the problem are comprised of a mix of several different ingredients, resulting in intricate algorithms. Moreover, the best of these algorithms is too complex to fully analyze analytically, and a computer was partly used in verifying its approximation factor. We propose a new approach to simplex partitioning and the multiway cut problem based on general transformations of the simplex that allow dependencies between the different variables. Our approach admits much simpler algorithms and, in addition, yields an approximation guarantee for the multiway cut problem that (roughly) matches the current best computer-verified approximation factor.