In this paper, we consider endomorphisms of a finite directed path from monoid generators perspective. Our main aim is to determine the rank [Formula: see text] all weak with vertices, which submonoid widely studied order-preserving transformations an text]-chain. Also, describe regular elements and calculate its size number idempotents.

We study the problem of answering k-hop reachability queries in a directed graph, i.e., whether there exists a directed path of length k, from a source query vertex to a target query vertex in the input graph. The problem of k-hop reachability is a general problem of the classic reachability (where k = ∞). Existing indexes for processing classic reachability queries, as well as for processing s...

An m-colored digraph is a digraph whose arcs are colored with m colors. A directed path is monochromatic when its arcs are colored alike. A set S ⊆ V (D) is a kernel by monochromatic paths whenever the two following conditions hold: 1. For any x, y ∈ S, x 6= y, there is no monochromatic directed path between them. 2. For each z ∈ (V (D)− S) there exists a zS-monochromatic directed path. In this...

We extend the notion of the Leavitt path algebra of a graph E to include all directed graphs. We show how various ring-theoretic properties of these more general structures relate to the corresponding properties of Leavitt path algebras of row-finite graphs. Specifically, we identify those graphs for which the corresponding Leavitt path algebra is simple; purely infinite simple; exchange; and s...

We study the problem of routing symmetric demand pairs in planar digraphs. The input consists of a directed planar graph G = (V, E) and a collection of k source-destination pairs M = {s1t1, . . . , sktk}. The goal is to maximize the number of pairs that are routed along disjoint paths. A pair siti is routed in the symmetric setting if there is a directed path connecting si to ti and a directed ...

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...

The technique of dynamic time warping (DTW) is relied on heavily in isolated word recognition systems. The advantage of using DTW is that reliable time alignment between reference and test patterns is obtained. The disadvantage of using DTW is the heavy computational burden required to find the optimal time alignment path. Several alternative procedures have been proposed for reducing the compu...

We call the digraph D an m-coloured digraph if the arcs of D 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) is said to be a kernel by monochromatic paths if it satisfies the two following conditions (i) for every pair of different vertices u, v ∈ N there is no monochromatic directed path between t...

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...