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...

Matrix M is k-concise if the finite entries of each column of M consist of k or fewer intervals of identical numbers. We give an O(n + m)-time algorithm to compute the row minima of any O(1)-concise n×m matrix. Our algorithm yields the first O(n+m)-time reductions from the replacement-paths problem on an n-node m-edge undirected graph (respectively, directed acyclic graph) to the single-source ...

We show that on a process graph defined by 1-semaphores, the space of directed paths (or dipaths) deformation retracts to a discrete set correlated by homotopy class relative to endpoints. This result extends to the set of ‘fastest executables’ (time-optimal dipaths). These results are obtained via techniques from cat(0) geometry and spaces of nonpositive curvature. keywords: CAT(0), nonpositiv...

In a previous work [26], by considering paths that are partially weighted, the generating function of Dyck paths was shown to possess a type of symmetry, called an exchange relation, derived from the exchange of a portion of the path between weighted and unweighted halves. This relation is particularly useful in solving for the generating functions of certain models of vertex-coloured Dyck path...