컨텐츠 시작
학술대회/행사
초록검색
| 제출번호(No.) | 0515 |
|---|---|
| 분류(Section) | Invited Talk |
| 분과(Session) | Combinatorics / Graph Theory / Cryptography / Coding Theory (SS-05) |
| 영문제목 (Title(Eng.)) |
Linear min-max relation between the treewidth of $H$-minor-free graphs and its largest grid minor |
| 저자(Author(s)) |
Ken-ichi Kawarabayashi1, Yusuke Kobayashi2 National Institute of Informatics1, University of Tokyo2 |
| 초록본문(Abstract) | A key theorem in algorithmic graph-minor theory is a min-max relation between the treewidth of a graph and its largest grid minor. This min-max relation is a keystone of the Graph Minor Theory of Robertson and Seymour, which ultimately proves Wagner's Conjecture about the structure of minor-closed graph properties. Demaine and Hajiaghayi proved a remarkable linear min-max relation for graphs excluding any fixed minor~$H$: every $H$-minor-free graph of treewidth at least $c_H r$ has an $r \times r$ grid minor for some constant $c_H$. However, as they pointed out, there is still a major problem left in this theorem. The problem is that their proof heavily depends on Graph Minor Theory, most of which lacks explicit bounds and is believed to have very large bounds. Hence $c_H$ is not explicitly given in their paper and therefore this result is usually not strong enough to derive efficient algorithms. Motivated by this problem, we give another (relatively short and simple) proof of this result without using big machinery of Graph Minor Theory. Hence we can give an explicit bound for $c_H$ (an exponential function of a polynomial of $|H|$). Furthermore, our result gives a constant $w=2^{O(r^2 \log r)}$ such that every graph of treewidth at least $w$ has an $r \times r$-grid minor, which improves the previously known best bound $2^{\Theta(r^5)}$ given by Robertson, Seymour, and Thomas in 1994. |
| 분류기호 (MSC number(s)) |
05C83, 05C85 |
| 키워드(Keyword(s)) | tree-width, grid minor, H-minor free graph |
| 강연 형태 (Language of Session (Talk)) |
English |