Adapting the Directed Grid Theorem into an FPT Algorithm

The Grid Theorem of Robertson and Seymour [JCTB, 1986], is one of the most important tools in the field of structural graph theory, finding numerous applications in the design of algorithms for undirected graphs. An analogous version of the Grid Theorem in digraphs was conjectured by Johnson et al. [JCTB, 2001], and proved by Kawarabayashi and Kreutzer [STOC, 2015]. Namely, they showed that there is a function $f(k)$ such that every digraph of directed tree-width at least $f(k)$ contains a cylindrical grid of size $k$ as a butterfly minor and stated that their proof can be turned into an XP algorithm, with parameter $k$, that either constructs a decomposition of the appropriate width, or finds the claimed large cylindrical grid as a butterfly minor. In this paper, we adapt some of the steps of the proof of Kawarabayashi and Kreutzer to improve this XP algorithm into an FPT algorithm. Towards this, our main technical contributions are two FPT algorithms with parameter $k$. The first one either produces an arboreal decomposition of width $3k-2$ or finds a haven of order $k$ in a digraph $D$, improving on the original result for arboreal decompositions by Johnson et al. The second algorithm finds a well-linked set of order $k$ in a digraph $D$ of large directed tree-width. As tools to prove these results, we show how to solve a generalized version of the problem of finding balanced separators for a given set of vertices $T$ in FPT time with parameter $|T|$, a result that we consider to be of its own interest.Comment: 31 pages, 9 figure

On disjoint directed cycles with prescribed minimum lengths

In this paper, we show that the k-Linkage problem is polynomial-time solvable for digraphs with circumference at most 2. We also show that the directed cycles of length at least 3 have the Erdős-Pósa Property : for every n, there exists an integer t_n such that for every digraph D, either D contains n disjoint directed cycles of length at least 3, or there is a set T of t_n vertices that meets every directed cycle of length at least 3. From these two results, we deduce that if F is the disjoint union of directed cycles of length at most 3, then one can decide in polynomial time if a digraph contains a subdivision of F

Complexity of greedy edge-colouring

The Grundy index of a graph G = (V,E) is the greatest number of colours that the greedy edge-colouring algorithm can use on G. We prove that the problem of determining the Grundy index of a graph G = (V,E) is NP-hard for general graphs. We also show that this problem is polynomial-time solvable for caterpillars. More specifically, we prove that the Grundy index of a caterpillar is $\Delta(G)$ or $\Delta(G)+1$ and present a polynomial-time algorithm to determine it exactly

Finding a subdivision of a digraph

International audienceWe consider the following problem for oriented graphs and digraphs: Given a directed graph D, does it contain a subdivision of a prescribed digraph F? We give a number of examples of polynomial instances, several NP-completeness proofs as well as a number of conjectures and open problems

Maximization Coloring Problems on graphs with few P4s

International audienceGiven a graph G = (V;E), a greedy coloring of G is a proper coloring such that, for each two colors i < j, every vertex of V(G) colored j has a neighbor with color i. The greatest k such that G has a greedy coloring with k colors is the Grundy number of G. A b-coloring of G is a proper coloring such that every color class contains a vertex which is adjacent to at least one vertex in every other color class. The greatest integer k for which there exists a b-coloring of G with k colors is its b-chromatic number. Determining the Grundy number and the b-chromatic number of a graph are NP-hard problems in general. For a fixed q, the (q;q-4)-graphs are the graphs for which no set of at most q vertices induces more than q-4 distinct induced P4s. In this paper, we obtain polynomial-time algorithms to determine the Grundy number and the b-chromatic number of (q;q-4)-graphs, for a fixed q. They generalize previous results obtained for cographs and P4-sparse graphs, classes strictly contained in the (q;q-4)-graphs

On finding the best and worst orientations for the metric dimension

The (directed) metric dimension of a digraph D, denoted by MD(D), is the size of a smallest subset S of vertices such that every two vertices of D are distinguished via their distances from the vertices in S. In this paper, we investigate the graph parameters BOMD(G) and WOMD(G) which are respectively the smallest and largest metric dimension over all orientations of G. We show that those parameters are related to several classical notions of graph theory and investigate the complexity of determining those parameters. We show that BOMD(G) = 1 if and only if G is hypotraceable (that is has a path spanning all vertices but one), and deduce that deciding whether BOMD(G) ≤ k is NP-complete for every positive integer k. We also show that WOMD(G) ≥ α(G) − 1, where α(G) is the stability number of G. We then deduce that for every fixed positive integer k, we can decide in polynomial time whether WOMD(G) ≤ k. The most significant results deal with oriented forests. We provide a linear-time algorithm to compute the metric dimension of an oriented forest and a linear-time algorithm that, given a forest F , computes an orientation D − with smallest metric dimension (i.e. such that MD(D −) = BOMD(F)) and an orientation D + with largest metric dimension (i.e. such that MD(D +) = WOMD(F))