On the treewidth of random geometric graphs and percolated grids

In this paper, we study the treewidth of the random geometric graph, obtained by dropping n points onto the square [0, √ n] and connect pairs of points by an edge if their distance is at most r = r(n). We prove a conjecture of Mitsche and Perarnau [19] stating that, with probability going to one as n→∞, the treewidth the random geometric graph is Θ(r √ n) when lim inf r > rc, where rc is the threshold radius for the appearance of the giant component. The proof makes use of a comparison to standard bond percolation and with a little bit of extra work we are also able to show that, with probability tending to one as k → ∞, the treewidth of graph we get by retaining each edge of the k × k-grid with probability p is Θ(k) if p > 1/2 and Θ( √ log k) if p < 1/2.