Doctor of Philosophy (PhD)
A rooted graph is a pair (G,R), where G is a graph and R⊆V(G). There are two research topics in this thesis. One is about unavoidable substructures in sufficiently large rooted graphs. The other is about characterizations of rooted graphs excluding specific large graphs.
The first topic of this thesis is motivated by Ramsey Theorem, which states that K_n and ¯(K_n ) are unavoidable induced subgraphs in every sufficiently large graph. It is also motivated by a classical result of Oporowski, Oxley, and Thomas, which determines unavoidable large 3-connected minors. We first determine unavoidable induced subgraphs, and unavoidable subgraphs in connected graphs with sufficiently many roots. We also extend this result to generalized rooted connected graphs. Secondly, we extend these results to rooted graphs of higher connectivity. In particular, we determine unavoidable subgraphs of sufficiently large rooted 2- connected graphs. Again, this result is extended to generalized rooted 2-connected graphs.
The second topic of this dissertation is motivated by two results of Robertson and Seymour, let’s only talk about path and star. In the first result they established that graphs without a long path subgraph are precisely those that can be constructed using a specific operation within a bounded number of iterations, starting from the trivial graph. In the second result they showed that graphs without a large star minor are those that are subdivisions of graphs with bounded number vertices. We consider similar problems for path, star and comb. We have some theorems on characterizations of rooted connected graphs excluding a heavy path, a large (nicely) confined comb, a large (nicely) confined star, which are similar to those of Robertson and Seymour. Moreover, our results strengthen their related results.
Wang, Shilin, "On Structures of Large Rooted Graphs" (2018). LSU Doctoral Dissertations. 4568.