Effective computation of immersion obstructions for unions of graph classes

• Immersions.
• Obstructions.
• Unique Linkage Theorem.
• Tree-width.

In the final paper of the Graph Minors series Robertson and Seymour proved that graphs are well-quasi-ordered under the immersion ordering. A direct implication of this theorem is that each class of graphs that is closed under taking immersions can be fully characterized by forbidding a finite set of graphs (immersion obstruction set). However, as the proof of the well-quasi-ordering theorem is non-constructive, there is no generic procedure for computing such a set. Moreover, it remains an open issue to identify for which immersion-closed graph classes the computation of those sets can become effective. By adapting the tools that were introduced by Adler, Grohe and Kreutzer, for the effective computation of minor obstruction sets, we expand the horizon of computability to immersion obstruction sets. In particular, our results propagate the computability of immersion obstruction sets of immersion-closed graph classes to immersion obstruction sets of finite unions of immersion-closed graph classes.