Aspects of random geometric graphs: Pursuit-evasion and treewidth

Abstract

In this thesis, we studied two aspects of random geometric graphs: pursuit-evasion and treewidth. We first studied one pursuit-evasion game: Cops and Robbers. This game, which dates back to 1970s, are studied extensively in recent years. We investigate this game on random geometric graphs, and get the number of cops needed to catch one Robber. To be more precise, we study the vertex pursuit game of Cops and Robbers, in which cops try to capture a robber on the vertices of a random geometric graph. The minimum number of cops required to win on a given graph G is called the cop number of G. We focus on G(n,r,p), a percolated random geometric graph in which n vertices are chosen uniformly at random and independently from [0,1]2, and two vertices are adjacent with probability p if the Euclidean distance between them is at most r. We present asymptotic results for the game of Cops and Robber played on G(n,r,p) for a wide range of p=p(n) and r=r(n). In the second part of my thesis, we considered the treewidth of two related random graph models. The first one is the random geometric graph, where we drop n points onto the square [0,√(n)]2 and connect pairs of points by an edge if their distance is at most r = r(n). The second one is the percolated grid, in which we take a k×k-grid and keep each edge with probability p, independently of all other edges. We show that, with probability tending to one as k→∞, the treewidth of the percolated grid is Θ(k) if p>1/2 and Θ(√(log k) ) if p < 1/2. By reusing part of the proof of this last result, we also prove a conjecture of Mitsche and Perarnan stating that, with probability going to one as n→∞, the treewidth the random geometric graph is Θ(r√(n)) when liminf r>rc, where rc is the threshold radius for the appearance of the giant component.