Georgia Institute of Technology College of Sciences

Graph immersion is an alternate model for graph containment similar to graph minors or topological minors. The presence of a large clique immersion in a graph G is closely related to the edge connectivity of G. This relationship gives rise to an easy theorem describing the structure of graphs excluding a fixed clique immersion which serves as the starting point for a broader structural theory of excluded immersions. We present the highlights of this theory with a look towards a conjecture of Nash-Williams on the well-quasi-ordering of graphs under strong immersions and a conjecture relating the chromatic number of a graph and the exclusion of a clique immersion.