Kernels for Feedback Arc Set In Tournaments

A tournament T=(V,A) is a directed graph in which there is exactly one arc between every pair of distinct vertices. Given a digraph on n vertices and an integer parameter k, the Feedback Arc Set problem asks whether the given digraph has a set of k arcs whose removal results in an acyclic digraph. The Feedback Arc Set problem restricted to tournaments is known as the k-Feedback Arc Set in Tournaments (k-FAST) problem. In this paper we obtain a linear vertex kernel for k-FAST. That is, we give a polynomial time algorithm which given an input instance T to k-FAST obtains an equivalent instance T' on O(k) vertices. In fact, given any fixed e>0, the kernelized instance has at most (2+e)k vertices. Our result improves the previous known bound of O(k^2) on the kernel size for k-FAST. Our kernelization algorithm solves the problem on a subclass of tournaments in polynomial time and uses a known polynomial time approximation scheme for k-FAST

Spanning a strong digraph by α circuits: A Proof of Gallai's Conjecture

In 1963, Tibor Gallai [9] asked whether every strongly connected directed graph D is spanned by α directed circuits, where α is the stability of D. We give a proof of this conjecture

Every strong digraph has a spanning strong subgraph with at most n + 2α - 2 arcs

Answering a question of Adrian Bondy [4], we prove that every strong digraph has a spanning strong subgraph with at most n + 2α − 2 arcs, where α is the size of a maximum stable set of D. Such a spanning subgraph can be found in polynomial time. An infinite family of oriented graphs for which this bound is sharp was given by Odile Favaron [3]. A direct corollary of our result is that there exists 2α − 1 directed cycles which span D. Tibor Gallai [6] conjectured that α directed cycles would be enough

The Categorical Product of two 5-chromatic digraphs can be 3-chromatic

We provide an example of a 5-chromatic oriented graph D such that the categorical product of D and TT5 is 3-chromatic, where TT5 is the transitive tournament on 5 vertices

Cyclic orderings and cyclic arboricity of matroids

We prove a general result concerning cyclic orderings of the elements of a matroid. For each matroid $M$, weight function $\omega:E(M)\rightarrow\mathbb{N}$, and positive integer $D$, the following are equivalent. (1) For all $A\subseteq E(M)$, we have $\sum_{a\in A}\omega(a)\le D\cdot r(A)$. (2) There is a map $\phi$ that assigns to each element $e$ of $E(M)$ a set $\phi(e)$ of $\omega(e)$ cyclically consecutive elements in the cycle $(1,2,...,D)$ so that each set $\{e|i\in\phi(e)\}$, for $i=1,...,D$, is independent. As a first corollary we obtain the following. For each matroid $M$ so that $|E(M)|$ and $r(M)$ are coprime, the following are equivalent. (1) For all non-empty $A\subseteq E(M)$, we have $|A|/r(A)\le|E(M)|/r(M)$. (2) There is a cyclic permutation of $E(M)$ in which all sets of $r(M)$ cyclically consecutive elements are bases of $M$. A second corollary is that the circular arboricity of a matroid is equal to its fractional arboricity. These results generalise classical results of Edmonds, Nash-Williams and Tutte on covering and packing matroids by bases and graphs by spanning trees