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...

it has been proved that sphericity testing for digraphs is an np-complete problem. here, we investigate sphericity of 3-connected single source digraphs. we provide a new combinatorial characterization of sphericity and give a linear time algorithm for sphericity testing. our algorithm tests whether a 3-connected single source digraph with $n$ vertices is spherical in $o(n)$ time.

A dicut in a directed graph is cut for which all of its edges are to common side the cut. famous theorem Lucchesi and Younger states that every finite digraph least size an edge set meeting equals maximum number disjoint dicuts digraph. In this first paper out series two papers, we conjecture version using more structural description min-max property infinite digraphs. We show can be reduced co...

For a graph G, the P2-path graph, P2(G), has for vertices the set of all paths of length 2 in G. Two vertices are connected when their union is a path or a cycle of length 3. We present lower bounds on the edge-connectivity, (P2(G)) of a connected graph G and give conditions for maximum connectivity. A maximally edge-connected graph is superif each minimum edge cut is trivial, and it is optimum...

Explicit expressions of the restricted edge connectivity of the Cartesian product of regular graphs are presented; some sufficient conditions for regular Cartesian product graphs to be maximally or super restricted edge connected are obtained as a result.

In this work, first, we present sufficient conditions for a bipartite digraph to attain optimum values of a stronger measure of connectivity, the so-called superconnectivity. To be more precise, we study the problem of disconnecting a maximally connected bipartite (di)graph by removing nontrivial subsets of vertices or edges. Within this framework, both an upper-bound on the diameter and Chartr...

If D = (V,A) is a digraph, its competition hypergraph CH(D) has vertex set V and e ⊆ V is an edge of CH(D) iff |e| ≥ 2 and there is a vertex v ∈ V , such that e = N− D (v) = {w ∈ V |(w, v) ∈ A}. We give characterizations of CH(D) in case of hamiltonian digraphs D and, more general, of digraphs D having a τ -cycle factor. The results are closely related to the corresponding investigations for co...

The main goal of this work is to establish a bijection between Dyck words and a family of Eulerian digraphs. We do so by providing two algorithms implementing such bijection in both directions. The connection between Dyck words and Eulerian digraphs exploits a novel combinatorial structure: a binary matrix, we call Dyck matrix, representing the cycles of an Eulerian digraph. 1. Background, and ...

The edge-connectivity of a graph G can be defined as λ(G) = min{λG(u, v) |u, v ∈ V (G)}, where λG(u, v) is the local edge-connectivity of two vertices u and v in G. We call a graph G maximally edgeconnected when λ(G) = δ(G) and maximally local edge-connected when λG(u, v) = min{d(u), d(v)} for all pairs u and v of distinct vertices in G. In 2000, Fricke, Oellermann and Swart (unpublished manusc...

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