KAIST Discrete Math Seminar

Graph immersion is a natural containment relation like graph minors. However, until recently, graph immersion has received relatively little attention. In this talk we shall describe some recent progress toward understanding when a graph does not immerse a certain subgraph. Namely, we detail a rough structure theorem for graphs which do not have K_t as an immersion, and we discuss the precise structure of graphs which do not have K_{3,3} as an immersion.

Then we turn our attention to a special class of digraphs, those for which every vertex has both indegree and outdegree equal to 2. These digraphs have special embeddings in surfaces where every vertex has a local rotation in which the inward and outward edges alternate. It turns out that the nature of these embeddings relative to immersion is quite closely related to the usual theory of graph embedding and graph minors. Here we describe the complete list of forbidden immersions for (special) embeddings in the projective plane.

These results are joint with various coauthors including Archdeacon, Dvorak, Fox, Hannie, Malekian, McDonald, Mohar, and Scheide.