Finding an anti-risk path between two nodes in undirected graphs

Given a weighted graph G = (V ,E) with a source s and a destination t , a traveler has to go from s to t . However, some of the edges may be blocked at certain times, and the traveler only observes that upon reaching an adjacent site of the blocked edge. Let P = {PG(s, t)} be the set of all paths from s to t . The risk of a path is defined as the longest travel under the assumption that any edge of the path may be blocked. The paper will propose the Anti-risk Path Problem of finding a path PG(s, t) in P such that it has minimum risk. We will show that this problem can be solved in O(mn+ n2 logn) time suppose that at most one edge may be blocked, where n and m denote the number of vertices and edges in G, respectively.