Twinless Strongly Connected Components

Abstract

Tarjan [9] describes how depth first search can be used to identify Strongly Connected Components (SCC) of a directed graph in linear time. It is standard to study Tarjan’s SCC algorithm in most senior undergraduate or introductory graduate computer science algorithms courses. In this paper we introduce the concept of a twinless strongly connected component (TSCC) of a directed graph. Loosely stated, a TSCC of a directed graph is (i) strongly connected, and (ii) remains strongly connected even if we require the deletion of arcs from the component, so that it does not contain a pair of twin arcs (twin arcs are a pair of bidirected arcs (i, j) and (j, i) where the tail of one arc is the head of the other and vice versa). This structure has diverse applications, from the design of telecommunication networks [7] to structural stability of buildings [8]. In this paper, we illustrate the relationship between 2-edge connected components of an undirected graph—obtained from the strongly connected components of a directed graph—and twinless strongly connected components. We use this relationship to develop a linear time algorithm to identify all the twinless strongly connected components of a directed graph. We then consider the augmentation problem, and based on the structural properties developed earlier, derive a linear time algorithm for the augmentation problem.