arXiv subjects
Publications and source records attributed to Henning Bruhn.
Menger's theorem on $A$--$B$-paths and Gallai's theorem on $A$-paths are among the most useful results in structural graph theory. Many variants and extensions are known. We add to this line of research and prove results that relate the maximal number of vertex-disjoint paths between vertex sets and a wall to the minimum number of vertices meeting all these paths. We also include types of paths that start and end in the wall.
We show that walls of size at least $6 \times 4$ do not have the edge-Erdős-Pósa property.
We show that subcubic graphs of treewidth at least $2500$ do not have the edge-Erdős-Pósa property.
We show that A-paths of length 0 modulo 4 have the Erdős-Pósa property. We also prove that A-paths of length 2 modulo 4 have the property but that A-paths of length 1 or of length 3 modulo 4 do not have it.
We optimise the production line of a manufacturing company in southern Germany in order to improve throughput. While the optimisation problem is NP-hard in general, analysing production data we find that in practice the problem can be solved very efficiently by aggressive generation of random machine schedules.
I prove that even $A$-cycles have the edge-Erdős-Pósa property.
In the 1960s, Erdős and Pósa proved that there is a packing-covering duality for cycles in graphs. As part of the graph minor project, Robertson and Seymour greatly extended this: there is such a duality for $H$-expansions in graphs if and only if $H$ is a planar graph (this includes the previous result for $H=K_3$). We consider vertex labelled graphs and minors and provide such a characterisation for $2$-connected labelled graphs $H$.
Robertson and Seymour proved that the family of all graphs containing a fixed graph $H$ as a minor has the Erdős-Pósa property if and only if $H$ is planar. We show that this is no longer true for the edge version of the Erdős-Pósa 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.
We prove that every graph $G$ contains either $k$ edge-disjoint $K_4$-subdivisions or a set $X$ of at most $O(k^8 \log k)$ edges such that $G-X$ does not contain any $K_4$-subdivision. This shows that $K_4$-subdivisions have the edge-Erdős-Pósa property.
We conjecture that any graph $G$ with treewidth~$k$ and maximum degree $Δ(G)\geq k + \sqrt{k}$ satisfies $χ'(G)=Δ(G)$. In support of the conjecture we prove its fractional version. We also show that any graph $G$ with treewidth~$k\geq 4$ and maximum degree $2k-1$ satisfies $χ'(G)=Δ(G)$, improving an old result of Vizing.
A key feature of Simonovits' proof of the classic Erdős-Pósa theorem is a simple subgraph of the host graph, a frame, that determines the outcome of the theorem. We transfer this frame technique to $A$-paths. With it we deduce a simple proof of Gallai's theorem, although with a worse bound, and we verify the Erdős-Pósa property for long and for even $A$-paths. We also show that even $A$-paths do not have the edge-Erdős-Pósa property.
We describe a fast approximation algorithm for the $Δ$-separated sparsity projection problem. The $Δ$-separated sparsity model was introduced by Hegde, Duarte and Cevher (2009) to capture the firing process of a single Poisson neuron with absolute refractoriness. The running time of our projection algorithm is linear for an arbitrary (but fixed) precision and it is both a head and a tail approximation. This solves a problem of Hegde, Indyk and Schmidt (2015). We also describe how our algorithm fits into the approximate model iterative hard tresholding framework of Hegde, Indyk and Schmidt (2014) that allows to recover $Δ$-separated sparse signals from noisy random linear measurements. The resulting recovery algorithm is substantially faster than the existing one, at least for large data sets.
We prove that $\det A\leq 6^\frac{n}{6}$ whenever $A\in\{0,1\}^{n\times n}$ contains at most $2n$ ones. We also prove an upper bound on the determinant of matrices with the $k$-consecutive ones property, a generalisation of the consecutive ones property, where each row is allowed to have up to $k$ blocks of ones. Finally, we prove an upper bound on the determinant of a path-edge incidence matrix in a tree and use that to bound the leaf rank of a graph in terms of its order.
We prove that the set of long cycles has the edge-Erdős-Pósa property: for every fixed integer $\ell\ge 3$ and every $k\in\mathbb{N}$, every graph $G$ either contains $k$ edge-disjoint cycles of length at least $\ell$ (long cycles) or an edge set $X$ of size $O(k^2\log k + \ell k)$ such that $G-X$ does not contain any long cycle. This answers a question of Birmelé, Bondy, and Reed (Combinatorica 27 (2007), 135--145).
A graph is called $t$-perfect if its stable set polytope is fully described by non-negativity, edge and odd-cycle constraints. We characterise $P_5$-free $t$-perfect graphs in terms of forbidden $t$-minors. Moreover, we show that $P_5$-free $t$-perfect graphs can always be coloured with three colours, and that they can be recognised in polynomial time.
We characterise $t$-perfect plane triangulations by forbidden induced subgraphs. As a consequence, we obtain that a plane triangulation is $h$-perfect if and only if it is perfect.
We prove that for every graph, any vertex subset $S$, and given integers $k,\ell$: there are $k$ disjoint cycles of length at least $\ell$ that each contain at least one vertex from $S$, or a vertex set of size $O(\ell \cdot k \log k)$ that meets all such cycles. This generalises previous results of Fiorini and Hendrickx and of Pontecorvi and Wollan. In addition, we describe an algorithm for our main result that runs in $O(k \log k \cdot s^2 \cdot (f(\ell) \cdot n+m))$ time, where $s$ denotes the cardinality of $S$.
We prove $χ_s'(G)\leq 1.93 Δ(G)^2$ for graphs of sufficiently large maximum degree where $χ_s'(G)$ is the strong chromatic index of $G$. This improves an old bound of Molloy and Reed. As a by-product, we present a Talagrand-type inequality where it is allowed to exclude unlikely bad outcomes that would otherwise render the inequality unusable.