Algorithm for k-truncated metric dimension of trees

The k-truncated metric dimension of a graph is the minimum number of sensors (a subset of the vertex set) needed to uniquely identify every vertex in the graph based on its distance to the sensors, where the sensors have a measuring range of k. We give an algorithm with the goal that given any tree and any value for the measuring range of the...

A graph theoretical approach to modelling growing cellular tissue

All organisms are built out of cellular tissue. Being able to recognise abnormalities in these tissues could be useful in recognizing cancerous cells. In this thesis we construct a mathematical model for cellular tissue based on its spatial structure. We consider cells as elements of the network. Touching cells are considered connected. Cells...

Small-world and ultrasmall-world phenomena in kernel-based spatial random graphs

In this thesis, we examine the kernel-based spatial random graph (KSRG) model, which is a generalisation of many known models such as long-range percolation, scale-free percolation, the Poisson Boolean model and age-based spatial preferential attachment. We construct a KSRG from a vertex set V = Z^d, assigning each vertex v ∈ V a weight Wv...

Effects of Fitness on Generative Graph Models with Preferential Attachment

Preferential Attachment models offer an explanation for why power laws are so common in real-world data. In these models, we start out with an initial network and add nodes one at a time. For each new node, we make m connections to existing nodes and if we define the attachment probability of attaching to a vertex to be proportional to its...

Extremal graphs for threshold metric dimension

In this thesis, we consider the threshold metric dimension problem of graphs, related to and motivated by source detection.<br/>We construct a graph G = (V,E) for a given set of sensors of size m: {s1, s2, ..., sm} and a range k &gt; 0. We want that each node v ∈ V has a unique combination of distances (dk (s1, v),dk (s2, v), ...,dk (sm, v)),...

K-core in random graphs

A graph G=(V,E) is a mathematical model for a network with vertex set V and edge set E. A Random Graph model is a probabilistic graph. A Random Geometric Graph is a Random Graph were each vertex has a location in a space χ. We compare the Erdos-Rényi random graph, G(n,p), to the Random Geometric Graph model, RGG(n,r) where, in general we use r=c...

On the domination number and cop number of Erdős–Rényi random graphs

We consider the game cops and robbers, which is a pursuit-evasion game played on a graph G. The cops and the robber take turns moving across the vertices of G, where the goal for the cops is to eventually catch the robber. Specifically, we study the cop number of G, i.e. the minimum number of cops that is needed to catch the robber on G. We...