On the Crossing Numbers of the Joining of a Specific Graph on Six Vertices with the Discrete Graph

In the paper, we extend known results concerning crossing numbers of join products of small graphs of order six with discrete graphs. The crossing number of the join product G ∗ + D n for the graph G ∗ on six vertices consists of one vertex which is adjacent with three non-consecutive vertices of the 5-cycle. The proofs were based on the idea of establishing minimum values of crossings between two different subgraphs that cross the edges of the graph G ∗ exactly once. These minimum symmetrical values are described in the individual symmetric tables.

Formalization and Discrete Modelling of Communication in the Digital Age by Using Graph Theory

Globalization is an important characteristic of the digital age which is based on the informatization of the society as a social-economical and science-technical process for changing the information environment while keeping the rights of citizens and organizations. The key features of the digital age are knowledge orientation, digital representation, virtual and innovative nature, integration and inter-network interactions, remote access to the information resources, economic and social cohesion, dynamic development, etc. The graph theory is a suitable apparatus for discrete presentation, formalization, and model investigation of the processes in the modern society because each state of a process could be presented as a node in a discrete graph with connections to other states. The chapter discusses application of the graph theory for a discrete formalization of the communication infrastructure and processes for remote access to information and network resources. An extension of the graph theory like apparatus of Petri nets is discussed and some examples for objects investigation are presented.

Determining crossing numbers of the join products of two specific graphs of order six with the discrete graph

The main aim of the paper is to give the crossing number of the join product G* + Dn for the connected graph G* of order six consisting of P4 + D1 and of one leaf incident with some inner vertex of the path P4 on four vertices, and where Dn consists of n isolated vertices. In the proofs, it will be extend the idea of the minimum numbers of crossings between two different subgraphs from the set of subgraphs which do not cross the edges of the graph G* onto the set of subgraphs by which the edges of G* are crossed exactly once. Due to the mentioned algebraic topological approach, we are able to extend known results concerning crossing numbers for join products of new graphs. The proofs are done with the help of software that generates all cyclic permutations for a given number k, and creates a new graph COG for calculating the distances between all (k-1)! vertices of the graph. Finally, by adding one edge to the graph G*, we are able to obtain the crossing number of the join product of one other graph with the discrete graph Dn.

ON THE CROSSING NUMBER OF THE JOIN OF THE WHEEL ON FIVE VERTICES WITH THE DISCRETE GRAPH

We give the crossing number of the join product $W_{4}+D_{n}$, where $W_{4}$ is the wheel on five vertices and $D_{n}$ consists of $n$ isolated vertices. The proof is based on calculating the minimum number of crossings between two different subgraphs from the set of subgraphs which do not cross the edges of the graph $W_{4}$ and from the set of subgraphs which cross the edges of $W_{4}$ exactly once.

Topology Attack and Defense for Graph Neural Networks: An Optimization Perspective

Graph neural networks (GNNs) which apply the deep neural networks to graph data have achieved significant performance for the task of semi-supervised node classification. However, only few work has addressed the adversarial robustness of GNNs. In this paper, we first present a novel gradient-based attack method that facilitates the difficulty of tackling discrete graph data. When comparing to current adversarial attacks on GNNs, the results show that by only perturbing a small number of edge perturbations, including addition and deletion, our optimization-based attack can lead to a noticeable decrease in classification performance. Moreover, leveraging our gradient-based attack, we propose the first optimization-based adversarial training for GNNs. Our method yields higher robustness against both different gradient based and greedy attack methods without sacrifice classification accuracy on original graph.

Unmanned Aerial Vehicle Surveillance Under Visibility and Dwell-Time Constraints

A framework is introduced for planning unmanned aerial vehicle (UAV) flight paths for visual surveillance of ground targets, each having particular viewing requirements. Specifically, the framework is designed for instances in which each target is associated with a set of imaging parameters, including a desired: (i) tilt angle, (ii) azimuth, with the option of a 360 deg view, and (iii) dwell-time. Tours are sought to image the targets, while minimizing both the total mission time and the time required to reach the initial target. An ϵ-constraint scalarization is used to pose the multi-objective problem as a constrained optimization, which, through careful discretization, can be approximated as a discrete graph-search. It is shown that, in many cases, this approximation is equivalent to a generalized traveling salesperson problem (GTSP). A heuristic procedure for solving the discrete approximation and recovering solutions to the full routing problem is presented and illustrated through numerical studies.