Half integral packing, Erdős-Posá-property and graph minors

• Authors: Ken-ichi Kawarabayashi
• Venue: SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
• Year: 2007

We consider the following problem: Given a graph G. Decide whether or not G has k half-integral packing of an H-minor for fixed k and H. Here, k half-integral packing of an H-minor is that there are k H-minors such that each vertex is used at most twice. Let us observe that if G contains H as a minor, then G has |H| disjoint trees such that after contracting all the trees, we can get an H. So k half-integral packing of an H-minor means that there are k|H| trees T1, 1, ..., T1,|H|, T2, 1, ..., Tk,|H| such that trees Ti, 1, ..., T1,|H| are disjoint for 1 ≤ i ≤ k, and after contracting all these trees Ti, 1, ..., Ti,|H|, we can get an H for 1 ≤ i ≤ k, but each vertex is used at most twice in these trees T1, 1, ..., T1,|H|, T2, 1, ..., Tk,|H|. This problem is motivated by the study of the half disjoint paths problem and unspilittable flow problem, e.g. Schrijver, Seymour and Winkler [45], Kleinberg [22], Srinivasan [49] and Kolliopoulos and Stein [23]. Let us observe that when k = 1, this case is exactly testing a given minor. So our problem includes Robertson-Seymour's result [36]. Also, it would not follow from the existence of kH-minors. Furthermore, the algorithm of the half disjoint paths problem by Kleinberg [22] does not give our algorithm. Hence, our setting is not directly comparable to Robertson and Seymour's result [36], and Kleinberg's result [22]. We prove that when H = K6 or K7, this problem is solvable in polynomial time. Our motivation for the K6-minor is that the structure of graphs with no K6-minors. Unfortunately, it is not known at all, although the structure of graphs without K5-minors is well known from 1930's [54]. Moreover, we prove that Erdős-Pósa property holds for k half-integral packing of a K6-minor (a K7-minor, respectively). More precisely, either G has K half-integral packing of a K6-minor (a K7-minor, respectively) or G has a vertex set T of order at most f(k) for some function of k such that G - T does not contain any K6-minor (K7-minor, respectively). This settles the conjecture of Thomas [50] for the K6-minor case and the K7-minor case. Finally, we shall address why our approach works for the K6-minor case and the K7-minor case, but does not work for the K8-minor case.