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We prove that every connected P5-free graph has cop number at most two, solving a conjecture of Sivaraman. In order to do so, we first prove that every connected P5-free graph G with independence number at least three contains a three-vertex induced path with vertices a-b-c in order, such that every neighbor of c is also adjacent to one of a,b.
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This content will become publicly available on January 1, 2026
Tree Independence Number IV. Even-hole-free graphs
We prove that the tree independence number of every even-hole-free graph is at most polylogarithmic in its number of vertices. More explicitly, we prove that there exists a constant c > 0 such that for every integer n > 1 every n-vertex even-hole-free graph has a tree decomposition where each bag has stability (independence) number at most clog10 n. This implies that the Maximum Weight Independent Set problem, as well as several other natural algorithmic problems that are known to be NP-hard in general, can be solved in quasipolynomial time if the input graph is even-hole-free. The quasi-polynomial complexity will remain the same even if the exponent of the logarithm is reduced to 1 (which would be asymptotically best possible).
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- Award ID(s):
- 2348219
- PAR ID:
- 10580822
- Publisher / Repository:
- Society for Industrial and Applied Mathematics - Symposium on Discrete Algorithms
- Date Published:
- ISBN:
- 978-1-61197-832-2
- Page Range / eLocation ID:
- 4444 to 4461
- Subject(s) / Keyword(s):
- Combinatorics (math.CO) Discrete Mathematics (cs.DM) Data Structures and Algorithms (cs.DS)
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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