MAXIMUM INDUCED SUBGRAPHS OF K-TREES WITH COMPONENTS OF ORDER 1 OR 2 Liu, Yunjiao 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. 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 Pelsmajer, Michael 2014 2014-07 Thesis application/pdf islandora:8026 http://hdl.handle.net/10560/3363 MATH / Applied Mathematics Illinois Institute of Technology en In Copyright http://rightsstatements.org/page/InC/1.0/ Restricted Access