This paper presents an algorithm for dividing a graph into triconnected components. The algorithm runs in O(V+E) time and space, where V is the number of vertices and E is the number of edges. It is both theoretically optimal and efficient in practice. The algorithm uses depth-first search to find triconnected components, which are subgraphs that remain connected after the removal of any two vertices. The algorithm is based on the concept of separation pairs, which are pairs of vertices that, when removed, disconnect the graph. The algorithm first identifies separation pairs and then uses them to split the graph into triconnected components. The algorithm is efficient and works for both directed and undirected graphs. The paper also provides a detailed description of the algorithm, including the use of depth-first search to find separation pairs and the use of auxiliary structures to determine the triconnected components. The algorithm is proven to be correct and efficient, and it is applicable to a wide range of graphs.This paper presents an algorithm for dividing a graph into triconnected components. The algorithm runs in O(V+E) time and space, where V is the number of vertices and E is the number of edges. It is both theoretically optimal and efficient in practice. The algorithm uses depth-first search to find triconnected components, which are subgraphs that remain connected after the removal of any two vertices. The algorithm is based on the concept of separation pairs, which are pairs of vertices that, when removed, disconnect the graph. The algorithm first identifies separation pairs and then uses them to split the graph into triconnected components. The algorithm is efficient and works for both directed and undirected graphs. The paper also provides a detailed description of the algorithm, including the use of depth-first search to find separation pairs and the use of auxiliary structures to determine the triconnected components. The algorithm is proven to be correct and efficient, and it is applicable to a wide range of graphs.