Fast approximation schemes for K3, 3-minor-free or K5-minor-free graphs
β Scribed by Mohammadtaghi Hajiaghayi; Naomi Nishimura; Prabhakar Ragde; Dimitrios M. Thilikos
- Publisher
- Elsevier Science
- Year
- 2001
- Tongue
- English
- Weight
- 227 KB
- Volume
- 10
- Category
- Article
- ISSN
- 1571-0653
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## Abstract Carsten Thomassen conjectured that every longest circuit in a 3βconnected graph has a chord. We prove the conjecture for graphs having no __K__~3,3~ minor, and consequently for planar graphs. Β© 2008 Wiley Periodicals, Inc. J Graph Theory 58: 293β298, 2008
We show that for any integer k G 2 and any n-vertex graph G without a K 3, 3 Ε½ . or K minor, one can compute k induced subgraphs of G with treewidth no more 5 Ε½ . Ε½. Ε½ Ε½ 2 .. than 3k y 4 respectively, 6 k y 7 in O kn respectively, O kn q n time such that each vertex of G appears in exactly k y 1 of
We show that an n-vertex bipartite K 3,3 -free graph with n 3 has at most 2n -4 edges and that an n-vertex bipartite K 5 -free graph with n 5 has at most 3n -9 edges. These bounds are also tight. We then use the bound on the number of edges in a K 3,3 -free graph to extend two known NC algorithms fo