arXiv subjects

Jean-Florent Raymond

Publications and source records attributed to Jean-Florent Raymond.

39 records · Page 3Linked to original sources

arXiv2015

Low Polynomial Exclusion of Planar Graph Patterns

The celebrated grid exclusion theorem states that for every $h$-vertex planar graph $H$, there is a constant $c_{h}$ such that if a graph $G$ does not contain $H$ as a minor then $G$ has treewidth at most $c_{h}$. We are looking for patterns of $H$ where this bound can become a low degree polynomial. We provide such bounds for the following parameterized graphs: the wheel ($c_{h}=O(h)$), the double wheel ($c_{h}=O(h^2\cdot \log^{2} h)$), any graph of pathwidth at most 2 ($c_{h}=O(h^{2})$), and the yurt graph ($c_{h}=O(h^{4})$).

math.CO↗

arXiv2015

An edge variant of the Erdős-Pósa property

For every $r\in \mathbb{N}$, we denote by $θ_{r}$ the multigraph with two vertices and $r$ parallel edges. Given a graph $G$, we say that a subgraph $H$ of $G$ is a model of $θ_{r}$ in $G$ if $H$ contains $θ_{r}$ as a contraction. We prove that the following edge variant of the Erd{\H o}s-P{ó}sa property holds for every $r\geq 2$: if $G$ is a graph and $k$ is a positive integer, then either $G$ contains a packing of $k$ mutually edge-disjoint models of $θ_{r}$, or it contains a set $S$ of $f_r(k)$ edges such that $G\setminus S$ has no $θ_{r}$-model, for both $f_r(k) = O(k^2r^3 \mathrm{polylog}~kr)$ and $f_r(k) = O(k^4r^2 \mathrm{polylog}~kr).$

math.CO↗

arXiv2013

Polynomial Gap Extensions of the Erdős-Pósa Theorem

Given a graph $H$, we denote by ${\cal M}(H)$ all graphs that can be contracted to $H$. The following extension of the Erdős-Pósa Theorem holds: for every $h$-vertex planar graph $H$, there exists a function $f_{H}$ such that every graph $G$, either contains $k$ disjoint copies of graphs in ${\cal M}(H)$, or contains a set of $f_{H}(k)$ vertices meeting every subgraph of $G$ that belongs in ${\cal M}(H)$. In this paper we prove that this is the case for every graph $H$ of pathwidth at most 2 and, in particular, that $f_{H}(k) = 2^{O(h^2)}\cdot k^{2}\cdot \log k$. As a main ingredient of the proof of our result, we show that for every graph $H$ on $h$ vertices and pathwidth at most 2, either $G$ contains $k$ disjoint copies of $H$ as a minor or the treewidth of $G$ is upper-bounded by $2^{O(h^2)}\cdot k^{2}\cdot \log k$. We finally prove that the exponential dependence on $h$ in these bounds can be avoided if $H=K_{2,r}$. In particular, we show that $f_{K_{2,r}}=O(r^2\cdot k^2)$

cs.DM↗