Temporalizing digraphs via linear-size balanced bi-trees

In a directed graph $D$ on vertex set $v_1,\dots ,v_n$, a \emph{forward arc} is an arc $v_iv_j$ where $i<j$. A pair $v_i,v_j$ is \emph{forward connected} if there is a directed path from $v_i$ to $v_j$ consisting of forward arcs. In the {\tt Forward Connected Pairs Problem} ({\tt FCPP}), the input is a strongly connected digraph $D$, and the output is the maximum number of forward connected pairs in some vertex enumeration of $D$. We show that {\tt FCPP} is in APX, as one can efficiently enumerate the vertices of $D$ in order to achieve a quadratic number of forward connected pairs. For this, we construct a linear size balanced bi-tree $T$ (an out-tree and an in-tree with same size which roots are identified). The existence of such a $T$ was left as an open problem motivated by the study of temporal paths in temporal networks. More precisely, $T$ can be constructed in quadratic time (in the number of vertices) and has size at least $n/3$. The algorithm involves a particular depth-first search tree (Left-DFS) of independent interest, and shows that every strongly connected directed graph has a balanced separator which is a circuit. Remarkably, in the request version {\tt RFCPP} of {\tt FCPP}, where the input is a strong digraph $D$ and a set of requests $R$ consisting of pairs $\{x_i,y_i\}$, there is no constant $c>0$ such that one can always find an enumeration realizing $c.|R|$ forward connected pairs $\{x_i,y_i\}$ (in either direction).Comment: 11 pages, 2 figur

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

The Categorical Product of Two 5-Chromatic Digraphs can be 3-Chromatic

International audienceWe provide an example of a 5-chromatic oriented graph $D$ such that the categorical product of $D$ and $TT_5$ is 3-chromatic, where $TT_5$ is the transitive tournament on 5 vertices

Spanning a strong digraph by α circuits: A Proof of Gallai&apos;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

Spanning a strong digraph by α\alpha circuits: A proof of Gallai's conjecture

International audienceIn 1963, Tibor Gallai~\cite{TG} asked whether every strongly connected directed graph $D$ is spanned by $\alpha$ directed circuits, where $\alpha$ 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

Partitioning a Graph into a Cycle and an Anticycle: A Proof of Lehel's Conjecture

International audienceWe prove that every graph $G$ has a vertex partition into a cycle and an anticyle (a cycle in the complement of $G$). Emptyset, singletons and edges are considered as cycles. This problem was posed by Lehel and shown to be true for very large graphs by \L uczak, Rödl and Szemerédi~\cite{LRS}, and more recently for large graphs by Allen~\cite{PA}