Transversals of subtree hypergraphs and the source location problem in hypergraphs

A hypergraph $H=(V,E)$ is a subtree hypergraph if there is a tree~$T$ on~$V$ such that each hyperedge of~$E$ induces a subtree of~$T$. Since the number of edges of a subtree hypergraph can be exponential in $n=|V|$, one can not always expect to be able to find a minimum size transversal in time polynomial in~$n$. In this paper, we show that if it is possible to decide if a set of vertices $W\subseteq V$ is a transversal in time~$S(n)$ (\,where $n=|V|$\,), then it is possible to find a minimum size transversal in~$O(n^3\,S(n))$. This result provides a polynomial algorithm for the Source Location Problem\,: a set of $(k,l)$-sources for a digraph $D=(V,A)$ is a subset~$S$ of~$V$ such that for any $v\in V$ there are~$k$ arc-disjoint paths that each join a vertex of~$S$ to~$v$ and~$l$ arc-disjoint paths that each join~$v$ to~$S$. The Source Location Problem is to find a minimum size set of $(k,l)$-sources. We show that this is a case of finding a transversal of a subtree hypergraph, and that in this case~$S(n)$ is polynomial

On Maltsev Digraphs

This is an Open Access article, first published by E-CJ on 25 February 2015.We study digraphs preserved by a Maltsev operation: Maltsev digraphs. We show that these digraphs retract either onto a directed path or to the disjoint union of directed cycles, showing in this way that the constraint satisfaction problem for Maltsev digraphs is in logspace, L. We then generalize results from Kazda (2011) to show that a Maltsev digraph is preserved not only by a majority operation, but by a class of other operations (e.g., minority, Pixley) and obtain a O(|VG|4)-time algorithm to recognize Maltsev digraphs. We also prove analogous results for digraphs preserved by conservative Maltsev operations which we use to establish that the list homomorphism problem for Maltsev digraphs is in L. We then give a polynomial time characterisation of Maltsev digraphs admitting a conservative 2-semilattice operation. Finally, we give a simple inductive construction of directed acyclic digraphs preserved by a Maltsev operation, and relate them with series parallel digraphs.Peer reviewedFinal Published versio

Digraph Colouring and Arc-Connectivity

The dichromatic number $\vec\chi(D)$ of a digraph $D$ is the minimum size of a partition of its vertices into acyclic induced subgraphs. We denote by $\lambda(D)$ the maximum local edge connectivity of a digraph $D$. Neumann-Lara proved that for every digraph $D$, $\vec\chi(D) \leq \lambda(D) + 1$. In this paper, we characterize the digraphs $D$ for which $\vec\chi(D) = \lambda(D) + 1$. This generalizes an analogue result for undirected graph proved by Stiebitz and Toft as well as the directed version of Brooks' Theorem proved by Mohar. Along the way, we introduce a generalization of Haj\'os join that gives a new way to construct families of dicritical digraphs that is of independent interest.Comment: 34 pages, 11 figure

Rank-based linkage I: triplet comparisons and oriented simplicial complexes

Rank-based linkage is a new tool for summarizing a collection $S$ of objects according to their relationships. These objects are not mapped to vectors, and ``similarity'' between objects need be neither numerical nor symmetrical. All an object needs to do is rank nearby objects by similarity to itself, using a Comparator which is transitive, but need not be consistent with any metric on the whole set. Call this a ranking system on $S$. Rank-based linkage is applied to the $K$-nearest neighbor digraph derived from a ranking system. Computations occur on a 2-dimensional abstract oriented simplicial complex whose faces are among the points, edges, and triangles of the line graph of the undirected $K$-nearest neighbor graph on $S$. In $|S| K^2$ steps it builds an edge-weighted linkage graph $(S, \mathcal{L}, \sigma)$ where $\sigma(\{x, y\})$ is called the in-sway between objects $x$ and $y$. Take $\mathcal{L}_t$ to be the links whose in-sway is at least $t$, and partition $S$ into components of the graph $(S, \mathcal{L}_t)$, for varying $t$. Rank-based linkage is a functor from a category of out-ordered digraphs to a category of partitioned sets, with the practical consequence that augmenting the set of objects in a rank-respectful way gives a fresh clustering which does not ``rip apart`` the previous one. The same holds for single linkage clustering in the metric space context, but not for typical optimization-based methods. Open combinatorial problems are presented in the last section.Comment: 37 pages, 12 figure