Let G = (V, E) be a graph with vertex set V of size n and edge set E of size m. A vertex v ∈ V is called a hinge vertex if the distance of any two vertices becomes longer after v is removed. A graph without hinge vertex is called a hinge-free graph. In general, a graph G is k-geodetically connected or k-GC for short if G can tolerate any k - 1 vertices failures without increasing the distance among all the remaining vertices. In this paper, we show that recognizing a graph G to be k-GC for the largest value of k can be solved in O(nm) time. In addition, more efficient algorithms for recognizing the k-GC property on some special graphs are presented. These include the O(n + m) time algorithms on strongly chordal graphs (if a strong elimination ordering is given), ptolemaic graphs, and interval graphs, and an O(n2) time algorithm on undirected path graphs (if a characteristic tree model is given). Moreover, we show that if the input graph G is not hinge-free then finding all hinge vertices of G can be solved in the same time complexity on the above classes of graphs.