Recently, the authors [Electron. J. Combin. 29 (2022), Paper No. 4.8] gave Ramsey-type results for path cover/partition number of graphs. In this paper, we continue research about them focusing on digraphs, and find a relationship between forbidden structures in digraphs. Let D be weakly connected digraph. A family P subdigraphs is called cover (resp. partition) if ⋃P∈PV(P)=V(D) ⋃˙P∈PV(P)=V(D))...

We introduce a novel harmonic analysis for functions defined on the vertices of strongly connected directed graph (digraph) which random walk operator is cornerstone. As first step, we consider set eigenvectors as non-orthogonal Fourier-type basis over digraphs. find frequency interpretation by linking variation obtained from their Dirichlet energy to real part associated eigenvalues. From this...

We give a fully dynamic single-source shortest paths data structure for planar weighted digraphs with O˜(n4/5) worst-case update time and O(log2⁡n) query time. Here, single can either change the graph by inserting or deleting an edge, reset source s of interest. All known non-trivial planarity-exploiting exact algorithms to date had polynomial then extend our approach, obtaining that maintain d...

A complete list of all connected arc-transitive asymmetric digraphs of in-valence and out-valence 2 on up to 1000 vertices is presented. As a byproduct, a complete list of all connected 4-valent graphs admitting a 12 -arc-transitive group of automorphisms on up to 1000 vertices is obtained. Several graph-theoretical properties of the elements of our census are calculated and discussed.

We study the half-integral k-Directed Disjoint Paths Problem ( 2kDDPP) in highly strongly connected digraphs. The integral kDDPP is NP-complete even when restricted to instances where k = 2, and the input graph is L-strongly connected, for any L ≥ 1. We show that when the integrality condition is relaxed to allow each vertex to be used in two paths, the problem becomes efficiently solvable in h...

A vertex-deleted unlabeled subdigraph of a digraph D is a card of D. A dacard specifies the degree triple (a, b, c) of the deleted vertex along with the card, where a and b are respectively the indegree and outdegree of v and c is the number of symmetric pairs of arcs (each pair considered as unordered edge) incident with v. The degree (triple) associated reconstruction number, drn(D), of a dig...

A new graph-theoretic proof of the uniqueness of noneven digraphs G(n) on n vertices with the maximum number M(n) of arcs will be discussed. This proof is then shown to be easily modiied to prove the second uniqueness result , namely, for n > 4; there is exactly one equivalence class of maximal strongly-connected noneven digraphs G(T 1 (n)) on n vertices with M(n) ? 1 arcs. There are no maximal...

The paper deals with topology-induced containment output feedback for ensuring multi-consensus of homogeneous linear systems evolving over a weakly connected communication digraph. Starting from the extension recent characterization multi-consensus, decentralized static enforcing is designed based on suitable network-induced decomposition; neighborhood state-observer proposed completing design....