Georgia Institute of Technology College of Sciences

Let $G$ be a graph containing 5 different vertices $a_0, a_1, a_2, b_1$ and $b_2$. We say that $(G,a_0,a_1,a_2,b_1,b_2)$ is feasible if $G$ contains disjoint connected subgraphs $G_1, G_2$, such that $\{a_0, a_1, a_2\}\subseteq V(G_1)$ and $\{b_1, b_2\}\subseteq V(G_2)$. We give a characterization for $(G,a_0,a_1,a_2,b_1,b_2)$ to be feasible, answering a question of Robertson and Seymour. This is joint work with Changong Li, Robin Thomas, and Xingxing Yu.In this talk, we will discuss the operations we will use to reduce $(G,a_0,a_1,a_2,b_1,b_2)$ to $(G',a_0',a_1',a_2',b_1',b_2')$ with $|V(G)|+|E(G)|>|V(G')|+E(G')$, such that $(G,a_0,a_1,a_2,b_1,b_2)$ is feasible iff $(G',a_0',a_1',a_2'b_1',b_2')$ is feasible.