Effective Computation of Immersion Obstructions for Unions of Graph Classes
In the final paper of the Graph Minors series [Neil Robertson and Paul D. Seymour. Graph minors XXIII. Nash-Williams’ immersion conjecture J. Comb. Theory, Ser. B, 100(2):181–205, 2010.], N. Robertson and P. Seymour proved that graphs are well-quasi-ordered with respect to the immersion relation. 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 where introduced in [Isolde Adler, Martin Grohe and Stephan Kreutzer. Computing excluded minors, SODA, 2008: 641-650.] for the effective computation of obstruction sets for the minor relation, we expand the horizon of the computability of obstruction sets for immersion-closed graph classes. In particular, we prove that there exists an algorithm that, given the immersion obstruction sets of two graph classes that are closed under taking immersions, outputs the immersion obstruction set of their union.
KeywordsImmersions Obstructions Unique Linkage Theorem Tree-width
Unable to display preview. Download preview PDF.
- 1.Adler, I., Grohe, M., Kreutzer, S.: Computing excluded minors. In: Teng, S.-H. (ed.) SODA, pp. 641–650. SIAM (2008)Google Scholar
- 6.DeVos, M., Dvořák, Z., Fox, J., McDonald, J., Mohar, B., Scheide, D.: Minimum degree condition forcing complete graph immersion. ArXiv e-prints (January 2011)Google Scholar
- 7.Enderton, H.B.: A mathematical introduction to logic. Academic Press (1972)Google Scholar
- 10.Grohe, M., Kawarabayashi, K.I., Marx, D., Wollan, P.: Finding topological subgraphs is fixed-parameter tractable. In: STOC, pp. 479–488 (2011)Google Scholar
- 11.Kawarabayashi, K.I., Wollan, P.: A shorter proof of the graph minor algorithm: the unique linkage theorem. In: Schulman, L.J. (ed.) STOC, pp. 687–694. ACM (2010)Google Scholar
- 13.Mendelson, E.: Introduction to mathematical logic, 3rd edn. Chapman and Hall (1987)Google Scholar
- 14.Robertson, N., Seymour, P.D.: Graph minors. XXII. Irrelevant vertices in linkage problems (to appear)Google Scholar