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 directed cycle is called quasi-monochromatic if with at most one exception 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 following t...

While standard persistent homology has been successful in extracting information from metric datasets, its applicability to more general data, e.g. directed networks, is hindered by its natural insensitivity to asymmetry. We extend a construction of homology of digraphs due to Grigoryan, Lin, Muranov and Yau to the persistent framework. The result, which we call persistent path homology or PPH,...

Let D be a directed graph of order n. An anti-directed Hamilton cycle H in D is a Hamilton cycle in the graph underlying D such that no pair of consecutive arcs in H form a directed path in D. We prove that if D is a directed graph with even order n and if the indegree and the outdegree of each vertex of D is at least 23n then D contains an anti-directed Hamilton cycle. This improves a bound of...

We present a randomized algorithm for reconstructing directed rooted trees of n nodes and node degree at most d, by asking at most O(dn log n) path queries. Each path query takes as input an origin node and a target node, and answers whether there is a directed path from the origin to the target. Our algorithm is only a factor of O(d log n) from the informationtheoretic lower bound of Ω(n log n...

This paper focuses on developing an efficient algorithm for analyzing a directed network (graph) from topological viewpoint. A prevalent technique such analysis involves computation of homology groups and their persistence. These concepts are well suited spaces that not directed. As result, one needs concept accommodates orientations in input space. Path-homology developed graphs by Grigoryan e...

The directed vertex leafage of a chordal graph G is the smallest integer k such that G is the intersection graph of subtrees of a rooted directed tree where each subtree has at most k leaves. In this note, we show how to find in time O(kn) an optimal colouring, a maximum independent set, a maximum clique, and an optimal clique cover of an n-vertex chordal graph G with directed vertex leafage k ...

Given a directed graph G and a parameter k, the Long Directed Cycle (LDC) problem asks whether G contains a simple cycle on at least k vertices, while the k-Path problems asks whether G contains a simple path on exactly k vertices. Given a deterministic (randomized) algorithm for k-Path as a black box, which runs in time t(G, k), we prove that LDC can be solved in deterministic time O(max{t(G, ...

A new algorithm, SPMF, for finding the complete chain of solutions of the product selection model is presented in this report. λ-directed simple residual path is identified to the only kind of residual path necessary for the new algorithm. By augmenting the right amount of flows along λ-directed simple residual paths, the new algorithm is monotone convergent. (Note: This work was first presente...

We study the max-min paths problem, which represents a game version of the shortest and the longest paths problem in a weighted directed graph. In this problem the vertex set V of the weighted directed graph G = (V, E) is divided into two disjoint subsets VA and VB which are regarded as positional sets of two players. The players are seeking for a directed path from the given starting position ...

A ranking of a graph is a coloring of the vertex set with positive integers such that on every path connecting two vertices of the same color there is a vertex of larger color. We consider the directed variant of this problem, where the above condition is imposed only on those paths in which all edges are oriented in the same direction. We show that the ranking number of a directed tree is boun...