On using distances to locate vertices: resolving sets and metric bases of graphs, two generalisations and their forced vertices

A graph consists of vertices that are connected by edges. A resolving set of a graph is a subset of its vertices that gives a unique combination of distances to every vertex of the graph. We can use the distances we are given to locate a vertex within the graph we are considering. Resolving sets were introduced by Slater in 1975 and independently by Harary and Melter in 1976. Robot navigation and network discovery and verification are examples of applications that have been suggested for resolving sets. In this dissertation, we consider resolving sets and two of their generalisations that can be used to locate subsets of vertices instead of individual vertices. We consider how these generalisations are connected to other concepts such as locatingdominating sets and the boundary of a graph. We place special emphasis on studying the minimum cardinalities of resolving sets and the two generalisations. In addition to proving general bounds to these minimum cardinalities, we consider their exact values in some graph families. Natural decision problems arise from some of the concepts that we consider and we study their algorithmic complexities. We also investigate which vertices of a graph must be included in an optimal resolving set or one of the two generalisations. For the resolving sets that can be used to locate subsets of vertices, there exist vertices that are in all such resolving sets. We call these vertices forced vertices of the graph. Such vertices do not exist for regular resolving sets. However, for minimum resolving sets they can exist, and we call them basis forced vertices of the graph. In this dissertation, we characterise the forced vertices of a graph, and consider some extremal properties of graphs that contain basis forced vertices

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

Distances and Domination in Graphs

This book presents a compendium of the 10 articles published in the recent Special Issue “Distance and Domination in Graphs”. The works appearing herein deal with several topics on graph theory that relate to the metric and dominating properties of graphs. The topics of the gathered publications deal with some new open lines of investigations that cover not only graphs, but also digraphs. Different variations in dominating sets or resolving sets are appearing, and a review on some networks’ curvatures is also present

Characterization of perfect Roman domination edge critical‎ ‎trees

‎A perfect Roman dominating function on a graph $G =(V‎, ‎E)$ is a function $f‎: ‎V \longrightarrow \{0‎, ‎1‎, ‎2\}$‎ ‎satisfying the condition that every vertex $u$ with $f(u) = 0$ is adjacent to exactly one vertex‎ ‎$v$ for which $f(v) = 2$‎. ‎The weight of a perfect Roman dominating function $f$ is the sum of‎ ‎the weights of the vertices‎. ‎The perfect Roman domination number of $G$‎, ‎denoted by $\gamma_{R}^{p}(G)$‎, ‎is‎ ‎the minimum weight of a perfect Roman dominating function in $G$‎. ‎In this paper‎, ‎we study the‎ ‎graphs for which adding any new edge decreases the perfect Roman‎ ‎domination number‎. ‎We call these graphs $\gamma_R^p$-edge critical‎. ‎The purpose of this paper is to characterize the class of‎ ‎$\gamma_R^p$-edge critical trees‎

Theoretical Computer Science and Discrete Mathematics

This book includes 15 articles published in the Special Issue "Theoretical Computer Science and Discrete Mathematics" of Symmetry (ISSN 2073-8994). This Special Issue is devoted to original and significant contributions to theoretical computer science and discrete mathematics. The aim was to bring together research papers linking different areas of discrete mathematics and theoretical computer science, as well as applications of discrete mathematics to other areas of science and technology. The Special Issue covers topics in discrete mathematics including (but not limited to) graph theory, cryptography, numerical semigroups, discrete optimization, algorithms, and complexity