In 1985, Mader conjectured the existence of a function f such that every digraph with minimum out-degree at least f (k) contains a subdivision of the transitive tournament of order k. This conjecture is still completely open, as the existence of f (5) remains unknown. In this paper, we show that if D is an oriented path, or an in-arborescence (i.e., a tree with all edges oriented towards the root) or the union of two directed paths from x to y and a directed path from y to x, then every digraph with minimum out-degree large enough contains a subdivision of D. Additionally, we study Mader's conjecture considering another graph parameter. The dichromatic number of a digraph D is the smallest integer k such that D can be partitioned into k acyclic subdigraphs. We show that any digraph with dichromatic number greater than 4 m (n − 1) contains every digraph with n vertices and m arcs as a subdivision.Conjecture 2 (Mader [20]). There exists a least integer mader δ + (T T k ) such that every digraph D with δ + (D) ≥ mader δ + (T T k ) contains a subdivision of T T k .Mader proved that mader δ + (T T 4 ) = 3, but even the existence of mader δ + (T T 5 ) is still open. This conjecture implies directly that transitive tournaments (and thus all acyclic digraphs) are δ 0maderian. Conjecture 3. There exists a least integer maderIn fact, Conjecture 3 is equivalent to Conjecture 2 because if transitive tournaments are δ 0 -maderian, then mader δ + (T T k ) ≤ mader δ 0 (T T 2k ) for all k. Indeed, let D be a digraph with minimum out-degree mader δ 0 (T T 2k ), and let D ′ be the digraph obtained from disjoint copies of D and its converse (the digraph obtained by reversing all arcs) D by adding all arcs from D to D. Clearly, δ 0 (D ′ ) ≥ mader δ 0 (T T 2k ). Therefore D ′ contains a subdivision of T T 2k . Hence, either D or D (and so D) contains a subdivision of T T k .Both conjectures are equivalent, but the above reasoning does not prove that a δ 0 -maderian digraph is also δ + -maderian. The case of oriented trees (i.e. orientations of undirected trees) is typical. Using a simple greedy procedure, one can easily find every oriented tree of order k in every digraph with minimum in-and out-degree k (so mader δ 0 (T ) = |T | − 1 for any oriented tree T ). On the other hand, it is still open whether oriented trees are δ + -maderian and a natural important step towards Conjecture 2 would be to prove the following weaker one.Conjecture 4. Every oriented tree is δ + -maderian.We give evidences to this conjecture. First, in Subsection 2.1, we prove that every oriented path (i.e. orientation of an undirected path) P is δ + -maderian and that mader δ + (P ) = |V (P )| − 1. Next, in Subsection 2.2, we consider arborescences. An out-arborescence (resp. in-arborescence) is an oriented tree in which all arcs are directed away from (resp. towards) a vertex called the root. Trivially, the simple greedy procedure shows that mader δ + (T ) = |T | − 1 for every out-arborescence. In contrast, the fact that in-arborecences are δ + -maderian is no...

An oriented cycle is an orientation of a undirected cycle. We first show that for any oriented cycle C, there are digraphs containing no subdivision of C (as a subdigraph) and arbitrarily large chromatic number. In contrast, we show that for any C a cycle with two blocks, every strongly connected digraph with sufficiently large chromatic number contains a subdivision of C. We prove a similar result for the antidirected cycle on four vertices (in which two vertices have out‐degree 2 and two vertices have in‐degree 2).

In this short note we prove that every tournament contains the k-th power of a directed path of linear length. This improves upon recent results of Yuster and of Girão. We also give a complete solution for this problem when k=2, showing that there is always a square of a directed path of length , which is best possible.

A very nice result of Bárány and Lehel asserts that every finite subset X or R d can be covered by f (d) X-boxes (i.e. each box has two antipodal points in X). As shown by Gyárfás and Pálvőlgyi this result would follow from the following conjecture : If a tournament admits a partition of its arc set into k quasi orders, then its domination number is bounded in terms of k. This question is in turn implied by the Erdős-Sands-Sauer-Woodrow conjecture : If the arcs of a tournament T are colored with k colors, there is a set X of at most g(k) vertices such that for every vertex v of T , there is a monochromatic path from X to v. We give a short proof of this statement. We moreover show that the general Sands-Sauer-Woodrow conjecture (which as a special case implies the stable marriage theorem) is valid for directed graphs with bounded stability number. This conjecture remains however open.

k-means and k-median clustering are powerful unsupervised machine learning techniques. However, due to complicated dependences on all the features, it is challenging to interpret the resulting cluster assignments. Moshkovitz, Dasgupta, Rashtchian, and Frost proposed an elegant model of explainable k-means and k-median clustering in ICML 2020. In this model, a decision tree with k leaves provides a straightforward characterization of the data set into clusters. We study two natural algorithmic questions about explainable clustering. (1) For a given clustering, how to find the ``best explanation'' by using a decision tree with k leaves? (2) For a given set of points, how to find a decision tree with k leaves minimizing the k-means/median objective of the resulting explainable clustering? To address the first question, we introduce a new model of explainable clustering. Our model, inspired by the notion of outliers in robust statistics, is the following. We are seeking a small number of points (outliers) whose removal makes the existing clustering well-explainable. For addressing the second question, we initiate the study of the model of Moshkovitz et al. from the perspective of multivariate complexity. Our rigorous algorithmic analysis sheds some light on the influence of parameters like the input size, dimension of the data, the number of outliers, the number of clusters, and the approximation ratio, on the computational complexity of explainable clustering.

The weak 2-linkage problem for digraphs asks for a given digraph and vertices s1, s2, t1, t2 whether D contains a pair of arc-disjoint paths P1, P2 such that Pi is an (si, ti)-path. This problem is NP-complete for general digraphs but polynomially solvable for acyclic digraphs [7]. Recently it was shown [9] that if D is equipped with a weight function w on the arcs which satisfies that all edges have positive weight, then there is a polynomial algorithm for the variant of the weak-2-linkage problem when both paths have to be shortest paths in D. In this paper we consider the unit weight case and prove that for every pair constants k1, k2, there is a polynomial algorithm which decides whether the input digraph D has a pair of arc-disjoint paths P1, P2 such that Pi is an (si, ti)-path and the length of Pi is no more than d(si, ti) + ki, for i = 1, 2, where d(si, ti) denotes the length of the shortest (si, ti)path. We prove that, unless the exponential time hypothesis (ETH) fails, there is no polynomial algorithm for deciding the existence of a solution P1, P2 to the weak 2-linkage problem where each path Pi has length at most d(si, ti) + c log 1+ǫ n for some constant c. We also prove that the weak 2-linkage problem remains NPcomplete if we require one of the two paths to be a shortest path while the other path has no restriction on the length.

The disjoint paths problem is a fundamental problem in algorithmic graph theory and combinatorial optimization. For a given graph G and a set of k pairs of terminals in G, it asks for the existence of k vertex-disjoint paths connecting each pair of terminals. The proof of Robertson and Seymour [JCTB 1995] of the existence of an n 3 algorithm for any fixed k is one of the highlights of their Graph Minors project. In this paper, we focus on the version of the problem where all the paths are required to be shortest paths. This problem, called the disjoint shortest paths problem, was introduced by Eilam-Tzoreff [DAM 1998] where she proved that the case k = 2 admits a polynomial time algorithm. This problem has received some attention lately, especially since the proof of the existence of a polynomial time algorithm in the directed case when k = 2 by Bérczi and Kobayashi [ESA 2017]. However, the existence of a polynomial algorithm when k = 3 in the undirected version remained open since 1998. In this paper we show that for any fixed k, the disjoint shortest paths problem admits a polynomial time algorithm. In fact for any fixed C, the algorithm can be extended to treat the case where each path connecting the pair (s, t) has length at most d(s, t) + C.

We prove the existence of a function h(k) such that every simple digraph with minimum outdegree greater than h(k) contains an immersion of the transitive tournament on k vertices. This solves a conjecture of Devos, McDonald, Mohar and Scheide.In this note, all digraphs are without loops. A digraph D is simple if there is at most one arc from x to y for any x, y ∈ V (D). Note that arcs in opposite directions are allowed. The multiplicity of a digraph D is the maximum number of parallel arcs in the same direction in D. We say that a digraph D contains an immersion of a digraph H if the vertices of H are mapped to distinct vertices of D, and the arcs of H are mapped to directed paths joining the corresponding pairs of vertices of D, in such a way that these paths are pairwise arc-disjoint. If the directed paths are vertex-disjoint, we say that D contains a subdivision of H.Understanding the necessary conditions for graphs to contain a subdivision of a clique is a very natural and well-studied question. One of the most important examples is the following result by Mader [6]:Theorem 1 ([6]). For every k ≥ 1, there exists an integer f (k) such that every graph with minimum degree greater than f (k) contains a subdivision of K k .