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Register a new userA tight ErdH{o}s-Posa function for wheel minors
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Added by
Jean-Florent Raymond
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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Let $W_t$ denote the wheel on $t+1$ vertices. We prove that for every integer $t geq 3$ there is a constant $c=c(t)$ such that for every integer $kgeq 1$ and every graph $G$, either $G$ has $k$ vertex-disjoint subgraphs each containing $W_t$ as minor, or there is a subset $X$ of at most $c k log k$ vertices such that $G-X$ has no $W_t$ minor. This is best possible, up to the value of $c$. We conjecture that the result remains true more generally if we replace $W_t$ with any fixed planar graph $H$.
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Robertson and Seymour proved that the family of all graphs containing a fixed graph $H$ as a minor has the ErdH{o}s-Posa property if and only if $H$ is planar. We show that this is no longer true for the edge version of the ErdH{o}s-Posa property, and indeed even fails when $H$ is an arbitrary subcubic tree of large pathwidth or a long ladder. This answers a question of Raymond, Sau and Thilikos.
A set of n points in the plane which are not all collinear defines at least n distinct lines. Chen and Chvatal conjectured in 2008 that a similar result can be achieved in the broader context of finite metric spaces. This conjecture remains open even for graph metrics. In this article we prove that graphs with no induced house nor induced cycle of length at least~5 verify the desired property. We focus on lines generated by vertices at distance at most 2, define a new notion of ``good pairs that might have application in larger families, and finally use a discharging technique to count lines in irreducible graphs.
A chordless cycle, or equivalently a hole, in a graph $G$ is an induced subgraph of $G$ which is a cycle of length at least $4$. We prove that the ErdH{o}s-Posa property holds for chordless cycles, which resolves the major open question concerning the ErdH{o}s-Posa property. Our proof for chordless cycles is constructive: in polynomial time, one can find either $k+1$ vertex-disjoint chordless cycles, or $c_1k^2 log k+c_2$ vertices hitting every chordless cycle for some constants $c_1$ and $c_2$. It immediately implies an approximation algorithm of factor $mathcal{O}(sf{opt}log {sf opt})$ for Chordal Vertex Deletion. We complement our main result by showing that chordless cycles of length at least $ell$ for any fixed $ellge 5$ do not have the ErdH{o}s-Posa property.
We prove that there exists a function $f:mathbb{N}rightarrow mathbb{R}$ such that every digraph $G$ contains either $k$ directed odd cycles where every vertex of $G$ is contained in at most two of them, or a vertex set $X$ of size at most $f(k)$ hitting all directed odd cycles. This extends the half-integral ErdH{o}s-Posa property of undirected odd cycles, proved by Reed [Mangoes and blueberries. Combinatorica 1999], to digraphs.
We prove that there exists a function $f(k)=mathcal{O}(k^2 log k)$ such that for every $C_4$-free graph $G$ and every $k in mathbb{N}$, $G$ either contains $k$ vertex-disjoint holes of length at least $6$, or a set $X$ of at most $f(k)$ vertices such that $G-X$ has no hole of length at least $6$. This answers a question of Kim and Kwon [ErdH{o}s-Posa property of chordless cycles and its applications. JCTB 2020].
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