A New Framework for Strong Connectivity and 2-Connectivity in Directed Graphs

In this paper, we investigate some basic problems related to the strong connectivity and to the 2-connectivity of a directed graph, by considering the effect of edge and vertex deletions on its strongly connected components. Let G be a directed graph with m edges and n vertices. We present a collection of O(n)-space data structures that, after O(m+ n)-time preprocessing can accomplish the following tasks: • Report in O(n) worst-case time all the strongly connected components obtained after deleting a single edge (resp., a single vertex) in G. • Compute the total number of strongly connected components obtained after deleting a single edge (resp., vertex) in total worst-case O(n) time for all edges (resp., for all vertices). • Compute the size of the largest and of the smallest strongly connected components obtained after deleting a single edge (resp., vertex) in total worst-case O(n) time for all edges (resp., for all vertices). All our bounds are tight. After the same O(m + n)-time preprocessing, we can also build an O(n)-space data structure that can answer in asymptotically optimal time the following basic 2-edge and 2-vertex connectivity queries on directed graphs: • Test whether an edge or a vertex separates two query vertices in O(1) worst-case time. • Report all edges or all vertices that separate two query vertices in optimal worst-case time. If there are k > 0 separating edges (resp., separating vertices), then the time is O(k) in the worst case. If there are no separating edges (resp., separating vertices), then the time is O(1) in the worst case. With the help of our data structures we can design efficient algorithms for several other strong connectivity problems on directed graphs. All our data structures are based on a uniform approach: in particular, we develop a common algorithmic framework, which hinges on new properties of the strongly connected components of a directed graph and on a novel combination of known graph theoretical concepts, such as dominance relations and loop nesting forests. Department of Computer Science & Engineering, University of Ioannina, Greece. E-mail: [email protected]. Dipartimento di Ingegneria Civile e Ingegneria Informatica, Università di Roma “Tor Vergata”, Roma, Italy. Email: {giuseppe.italiano, nikos.parotsidis}@uniroma2.it. Partially supported by MIUR, the Italian Ministry of Education, University and Research, under Project AMANDA (Algorithmics for MAssive and Networked DAta). 1 ar X iv :1 51 1. 02 91 3v 1 [ cs .D S] 9 N ov 2 01 5