Distributed Local 2-Connectivity Test of Graphs and Applications

Abstract

The vertex connectivity of a graph is the smallest number of vertices whose deletion separates the graph or makes it trivial. This work is devoted to the problem of vertex connectivity test of graphs in a distributed environment based on a constructive approach. The contribution of this paper is threefold. First, using a pre-constructed spanning tree of the considered graph, we present a protocol to test whether a given graph is 2-connected using only local knowledge. Second, we present an encoding of this protocol using graph relabeling systems. The last contribution is the implementation of this protocol in the message passing model. For a given graph $\: G ,\:$ where $\: M \:$ is the number of its edges, $\: N \:$ the number of its nodes and $\: \Delta \:$ is its degree, our algorithms need the following requirements: The first one uses $\:O(\Delta \times N^2 )\:$ steps and $\:O(\Delta \times \log \Delta)\:$ bits per node. The second one uses $\: O(\Delta \times N^2 )\:$ messages and $\:O(N^2)\:$ time and $\:O(\Delta \times \log \Delta)\:$ bits per node. Furthermore, the studied network is semi-anonymous: Only the root of the pre-constructed spanning tree needs to be identified. Moreover, we investigate some applications that can use our protocol as a pre-processing task for initial configurations.