We study induced subgraphs where every component has order 1 or 2. For a graph G, let f(G) be the maximum order of such a subgraph of G.... Show moreWe study induced subgraphs where every component has order 1 or 2. For a graph G, let f(G) be the maximum order of such a subgraph of G. Chappell and Pelsmajer [1] considered a more general parameter for graphs G of bounded treewidth, but were unable to determine f(G) for graphs of treewidth k > 3, even asymptotically. They conjectured that f(G) ≥ ⌈ 2n k+2⌉ for an n-vertex graph of treewidth at most k, but for k > 3, they were only able to show that f(G) ≥ 2n+2 2k+3 . In this thesis, we improve the lower bound to l 8n 5(k+1)m, for n ≥ 2k + 1. In addition, for the case k = 4, we develop methods for an inductive proof, where the cases are verified by computer-checking. If the conjecture is false, then our approach should eventually lead to a counter-example. To facilitate this approach, we come up with the addition structure on 4-trees, where one K4-subgraph is the “root”, and we consider all the different ways that an induced subgraph can intersect with the root separately. M.S. in Applied Mathematics, July 2014 Show less
