A Simple Algorithm For Triconnectivity of a Multigraph

Vertex-connectivity and edge-connectivity represent the extent to which a graph is connected. Study of these key properties of graphs plays an important role in varieties of computer science applications. Recent years have witnessed a number of linear time 3-edge-connectivity algorithms with increasing simplicity. In contrast, the state-of-the-art algorithm for 3-vertex-connectivity due to Hopcroft and Tarjan lacks the simplicity in the sense of ease of implementation as well as the number of passes over the graph although its time and space complexity is theoretically linear. In this paper, we propose a linear time reduction from 3-vertex-connectivity to 3-edgeconnectivity of a multigraph. This reduction was previously unknown, while the reduction in the opposite direction already exists. We apply an existing linear time 3-edge-connectivity algorithm on the reduced graph for solving the 3-vertex-connectivity of the original graph. Hence, for a graph with V vertices and E edges, the proposed reduction turns into an O(|V | +|E|) time and space algorithm for 3-vertex-connectivity while enjoying the simplicity of the 3-edge-connectivity algorithms. Type of Report: Other Department of Computer Science & Engineering Washington University in St. Louis Campus Box 1045 St. Louis, MO 63130 ph: (314) 935-6160 Reduction From 3-Vertex-Connectivity to 3-Edge-Connectivity Abusayeed M. Saifullah and Alper Üngör 1 Computer Science & Engineering Washington University, St. Louis, MO 63130, USA [email protected] 2 Computer & Info. Science & Engineering University of Florida, Gainesville, FL 32611, USA [email protected] Abstract. Vertex-connectivity and edge-connectivity represent the extent to which a graph is connected. Study of these key properties of graphs plays an important role in varieties of computer science applications. Recent years have witnessed a number of linear time 3-edgeconnectivity algorithms with increasing simplicity. In contrast, the state-of-the-art algorithm for 3-vertex-connectivity due to Hopcroft and Tarjan lacks the simplicity in the sense of ease of implementation as well as the number of passes over the graph although its time and space complexity is theoretically linear. In this paper, we propose a linear time reduction from 3vertex-connectivity to 3-edge-connectivity of a multigraph. This reduction was previously unknown, while the reduction in the opposite direction already exists. We apply an existing linear time 3-edge-connectivity algorithm on the reduced graph for solving the 3-vertex-connectivity problem of the original graph. Hence, for a graph with |V | vertices and |E| edges, the proposed reduction turns into an O(|V |+ |E|) time and space algorithm for 3-vertex-connectivity while enjoying the simplicity of the 3-edge-connectivity algorithms. Vertex-connectivity and edge-connectivity represent the extent to which a graph is connected. Study of these key properties of graphs plays an important role in varieties of computer science applications. Recent years have witnessed a number of linear time 3-edgeconnectivity algorithms with increasing simplicity. In contrast, the state-of-the-art algorithm for 3-vertex-connectivity due to Hopcroft and Tarjan lacks the simplicity in the sense of ease of implementation as well as the number of passes over the graph although its time and space complexity is theoretically linear. In this paper, we propose a linear time reduction from 3vertex-connectivity to 3-edge-connectivity of a multigraph. This reduction was previously unknown, while the reduction in the opposite direction already exists. We apply an existing linear time 3-edge-connectivity algorithm on the reduced graph for solving the 3-vertex-connectivity problem of the original graph. Hence, for a graph with |V | vertices and |E| edges, the proposed reduction turns into an O(|V |+ |E|) time and space algorithm for 3-vertex-connectivity while enjoying the simplicity of the 3-edge-connectivity algorithms.