The chromatic number → χ (D) of a digraph D is the minimum number of colors needed to color the vertices of D such that each color class induces an acyclic sub-We examine methods for creating infinite families of critical digraphs, the Dirac join and the directed and bidirected Hajós join. We prove that a digraph D has chromatic number at least k if and only if it contains a subdigraph that can be obtained from bidirected complete graphs on k vertices by (directed) Hajós joins and identifying non-adjacent vertices. Building upon that, we show that a digraph D has chromatic number at least k if and only if it can be constructed from bidirected K k 's by using directed and bidirected Hajós joins and identifying non-adjacent vertices (so called Ore joins), thereby transferring a well-known result of Urquhart to digraphs. Finally, we prove a Gallai-type theorem that characterizes the structure of the low vertex subdigraph of a critical digraph, that is, the subdigraph, which is induced by the vertices that have in-degree k − 1 and out-degree k − 1 in D.
This paper studies the problem of traffic monitoring which consists of differentiating a set of walks on a directed graph by placing sensors on as few arcs as possible. The problem of characterising a set of individuals by testing as few attributes as possible is already well-known, but traffic monitoring presents new challenges that the previous models of separation fall short from modelling such as taking into account the multiplicity and order of the arcs in a walk. We introduce a new and stronger model of separation based on languages that generalises the traffic monitoring problem. We study three subproblems with practical applications and develop methods to solve them by combining integer linear programming, separating codes and language theory.
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 dichromatic number χ(D) of a digraph D is the least integer k such that D can be partitionedAn oriented graph is a digraph with no directed cycle of length 2. For integers k and n, we denote by o k (n) the minimum number of edges of a k-critical oriented graph on n vertices (with the convention o k (n) = +∞ if there is no k-dicritical oriented graph of order n). The main result of this paper is a proof that o 3 (n) ≥ 7n+2 3 together with a construction witnessing that o 3 (n) ≤ 5n 2 for all n ≥ 12. We also give a construction showing that for all sufficiently large n and all k ≥ 3, o k (n) < (2k − 3)n, disproving a conjecture of Hoshino and Kawarabayashi. Finally, we prove that, for all k ≥ 2, o k (n) ≥ k − 3 4 − 1 4k−6 n + 3 4(2k−3) , improving the previous best known lower bound of Bang-Jensen, Bellitto, Schweser and Stiebitz.
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