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 G ( 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 P G ( s , t ) in ℘ such that it has minimum risk. We will show that this problem can be solved in O ( mn + n 2 log n ) time suppose that at most one edge may be blocked, where n and m denote the number of vertices and edges in G , respectively.