A tight Erd\H{o}s-P\'osa function for wheel minors

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 $k\geq 1$ 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 $H$.Comment: 15 pages, 1 figur

On the Erd\H{o}s-P\'osa property for immersions and topological minors in tournaments

We consider the Erd\H{o}s-P\'osa property for immersions and topological minors in tournaments. We prove that for every simple digraph $H$, $k\in \mathbb{N}$, and tournament $T$, the following statements hold: $\bullet$ If in $T$ one cannot find $k$ arc-disjoint immersion copies of $H$, then there exists a set of $\mathcal{O}_H(k^3)$ arcs that intersects all immersion copies of $H$ in $T$. $\bullet$ If in $T$ one cannot find $k$ vertex-disjoint topological minor copies of $H$, then there exists a set of $\mathcal{O}_H(k\log k)$ vertices that intersects all topological minor copies of $H$ in $T$. This improves the results of Raymond [DMTCS '18], who proved similar statements under the assumption that $H$ is strongly connected.Comment: 15 pages, 1 figur

Approximate min-max relations on plane graphs

Let G be a plane graph, let τ(G) (resp. τ′(G)) be the minimum number of vertices (resp. edges) that meet all cycles of G, and let ν(G) (resp. ν′(G)) be the maximum number of vertex-disjoint (resp. edge-disjoint) cycles in G. In this note we show that τ(G)≤3 ν(G) and τ′(G)≤4 ν′(G)-1; our proofs are constructive, which yield polynomial-time algorithms for finding corresponding objects with the desired properties. © 2011 The Author(s).published_or_final_versionSpringer Open Choice, 28 May 201

On disjoint directed cycles with prescribed minimum lengths

In this paper, we show that the k-Linkage problem is polynomial-time solvable for digraphs with circumference at most 2. We also show that the directed cycles of length at least 3 have the Erdős-Pósa Property : for every n, there exists an integer t_n such that for every digraph D, either D contains n disjoint directed cycles of length at least 3, or there is a set T of t_n vertices that meets every directed cycle of length at least 3. From these two results, we deduce that if F is the disjoint union of directed cycles of length at most 3, then one can decide in polynomial time if a digraph contains a subdivision of F

On the odd-minor variant of Hadwiger's conjecture

A {\it $K_l$ -expansion} consists of $l$ vertex-disjoint trees, every two of which are joined by an edge. We call such an expansion {\it odd} if its vertices can be two-coloured so that the edges of the trees are bichromatic but the edges between trees are monochromatic. We show that, for every $l$, if a graph contains no odd $K_l$ -expansion then its chromatic number is $O(l \sqrt{\log l})$. In doing so, we obtain a characterization of graphs which contain no odd $K_l$ -expansion which is of independent interest. We also prove that given a graph and a subset $S$ of its vertex set, either there are $k$ vertex-disjoint odd paths with endpoints in $S$, or there is a set X of at most $2k − 2$ vertices such that every odd path with both ends in $S$ contains a vertex in $X$. Finally, we discuss the algorithmic implications of these results

Odd-Minors I: Excluding small parity breaks

Given a graph class~$\mathcal{C}$, the $\mathcal{C}$-blind-treewidth of a graph~$G$ is the smallest integer~$k$ such that~$G$ has a tree-decomposition where every bag whose torso does not belong to~$\mathcal{C}$ has size at most~$k$. In this paper we focus on the class~$\mathcal{B}$ of bipartite graphs and the class~$\mathcal{P}$ of planar graphs together with the odd-minor relation. For each of the two parameters, $\mathcal{B}$-blind-treewidth and ${(\mathcal{B}\cup\mathcal{P})}$-blind-treewidth, we prove an analogue of the celebrated Grid Theorem under the odd-minor relation. As a consequence we obtain FPT-approximation algorithms for both parameters. We then provide FPT-algorithms for \textsc{Maximum Independent Set} on graphs of bounded $\mathcal{B}$-blind-treewidth and \textsc{Maximum Cut} on graphs of bounded ${(\mathcal{B}\cup\mathcal{P})}$-blind-treewidth

On the odd-minor variant of Hadwiger's conjecture

A {\it $K_l$ -expansion} consists of $l$ vertex-disjoint trees, every two of which are joined by an edge. We call such an expansion {\it odd} if its vertices can be two-coloured so that the edges of the trees are bichromatic but the edges between trees are monochromatic. We show that, for every $l$, if a graph contains no odd $K_l$ -expansion then its chromatic number is $O(l \sqrt{\log l})$. In doing so, we obtain a characterization of graphs which contain no odd $K_l$ -expansion which is of independent interest. We also prove that given a graph and a subset $S$ of its vertex set, either there are $k$ vertex-disjoint odd paths with endpoints in $S$, or there is a set X of at most $2k − 2$ vertices such that every odd path with both ends in $S$ contains a vertex in $X$. Finally, we discuss the algorithmic implications of these results