Question

Suppose we are given a directed acyclic graph G with a unique source and a unique sink t. A vertex v ¢ {s,t} is called an (s,t)-cut vertex if every path from s to t passes through v, or equivalently, if deleting v makes t unreachable from s. Describe and analyze an algorithm to find every (s, t)-cut vertex in G t

Answer 1

Vertex t will only be reachable from vertex s if performing BFS(Breadth First Search) starting from vertex s will cover vertex t otherwise t will not be reachable from s.

Hence our algorithm will be as follows:-

CUT_VERTEX(G=(V,E),s,t)

1. Cut_vertex_set = {} //initially this set is empty

2. For each vertex $v \notin \{s,t\}$ in V:-

3..........Remove v from G

4..........Perform BFS in G with s as source vertex

5.................If BFS tree contains t then add vertex v back to graph G and continue //since v does not disconnect s and t

6................else //v is the cut vertex

7..........................Cut_vertex_set.Add(v) //add v into the solution set

8..........................Add vertex v back to graph G and continue

9. Return Cut_vertex_set //return the solution

Time complexity = |V|*( Time to remove vertex from G and adding back the vertex + Time to perform BFS)

=O( |V|*(|V|+|E| + |V|+|E|)) = O(|V|² + |V||E|)

Please comment for any clarification