Page 2 Lecture Notes in Computer Science 2865 Edited by G. Goos ...

262 S. Bhadra and A. FerreiraIn this paper, we start by providing, in the next section, basic definitions forvarious common graph theory terms in the context of evolving digraphs. FollowingHumblet [12], we define rooted DMST’s over strongly connected evolvingdigraphs. This naturally leads to the question of how to determine if an evolvingdigraph is strongly connected. In Section 3 we define strongly connected components(SCC’s) in evolving digraphs and discover that the unique propertiesof evolving digraphs yield two types of strongly connected components: standardSCC’s and the more loosely defined open strongly connected components(o-SCC’s), as it will become clear later. One of our results is that unlike in standarddigraphs, finding the strongly connected components in evolving digraphsis not possible in deterministic polynomial time, unless P=NP. In case the evolvingdigraph is already identified as a strongly connected component, we give inSection 4 an algorithm to compute DMST, which uses a variation of Prim’s algorithm[4] for computing minimum spanning trees. For an evolving digraph withmaximum outdegree D, our algorithm builds the rooted DMST over a stronglyconnected component in an evolving digraph in O(ND log T ) time. Section 5contains concluding remarks and scope for further research.2 Graph Theoretic ModelSince we use evolving digraphs as a model for FSDN’s throughout this paper,we start with a revision of the basic definitions of terms in the theory of evolvingdigraphs.2.1 Evolving DigraphsEvolving digraphs are defined as follows.Definition 1 (Evolving Digraphs). Let a digraph G(V,E) be given, alongwith an ordered sequence of its subdigraphs, S G = G 0 ,G 1 ,...,G T , T ∈ IN. Then,the system G =(G, S G ) is called an evolving digraph.We now define some of the main parameters of an evolving digraph. Let E G =⋃Ei , and V G = ⋃ V i . It is clear that M = |E G |≤|E| = M and that N = |V G |≤|V | = N. The central notion in evolving graph theory is the restriction imposedupon paths to traverse arcs strictly in non-decreasing order of arc schedule times,implying that there are no paths in G going to the “past.”Definition 2 (Journeys). Let P be a path in G i , under the usual definition.Let F (P ) be its first vertex, L(P ) be its last vertex, and |P | be its length. Wedefine a journey in G between two vertices u and v of V G as a sequence J (u, v) =P t1 ,P t2 ,...,P tk , with t 1