A digraph D is said to be an m-coloured digraph, if its arcs are coloured with m colours. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike. A set N ⊆ V (D) of vertices of D is said to be a kernel by monochromatic paths of the m-coloured digraph D, if it satisfies the two following properties: (1) N is independent by monochromatic paths; i.e., f...

We study Markov processes where the “time” parameter is replaced by paths in a directed graph from an initial vertex to a terminal one. Along each directed path the process is Markov and has the same distribution as the one along any other directed path. If two directed paths do not interact, in a suitable sense, then the distributions of the processes on the two paths are conditionally indepen...

A simplex spanned by a colored point set in Euclidean d-space is colorful if all vertices have distinct colors. The union of all full-dimensional colorful simplices spanned by a colored point set is called the colorful union. We show that for every d ∈ N, the maximum combinatorial complexity of the colorful union of n colored points in R is between Ω(n(d−1) 2 ) and O(n(d−1) 2 log n). For d = 2,...

The colorful Carathéodory theorem, proved by Bárány in 1982, states that given d+1 sets of points S1, . . . ,Sd+1 in R , such that each Si contains 0 in its convex hull, there exists a set T ⊆ ⋃ d+1 i=1 Si containing 0 in its convex hull and such that |T ∩Si| ≤ 1 for all i ∈ {1, . . . , d + 1}. An intriguing question – still open – is whether such a set T , whose existence is ensured, can be fo...

The NP-hard Colorful Components problem is a graph partitioning problem on vertex-colored graphs. We identify a new application of Colorful Components in the correction of Wikipedia interlanguage links, and describe and compare three exact and two heuristic approaches. In particular, we devise two ILP formulations, one based on Hitting Set and one based on Clique Partition. Furthermore, we use ...

Given k + 1 pairs of vertices (s1, s2), (u1, v1), . . . , (uk, vk) of a directed acyclic graph, we show that a modified version of a data structure of Suurballe and Tarjan can output, for each pair (ul, vl) with 1 ≤ l ≤ k, a tuple (s1, t1, s2, t2) with {t1, t2} = {ul, vl} in constant time such that there are two disjoint paths p1, from s1 to t1, and p2, from s2 to t2, if such a tuple exists. Di...

Semi-directed 2-circulant graph is a subgraph of an (undirected) 2-circulant graph in which the links of one type (i.e., short or long) are directed while the other links are undirected. The shortest paths in semi-directed circulant graphs are called the restricted shortest paths in 2-circulant graphs. In this paper we show that the problem of finding the restricted shortest paths is equivalent...

This paper develops a uni(ed enumerative and asymptotic theory of directed two-dimensional lattice paths in half-planes and quarter-planes. The lattice paths are speci(ed by a (nite set of rules that are both time and space homogeneous, and have a privileged direction of increase. (They are then essentially one-dimensional objects.) The theory relies on a speci(c “kernel method” that provides a...

The enumeration of lattice paths in wedges poses unique mathematical challenges. These models are not translationally invariant, and the absence of this symmetry complicates both the derivation of a functional recurrence for the generating function, and solving for it. In this paper we consider a model of partially directed walks from the origin in the square lattice confined to both a symmetri...