We introduce graphs called hypochordal: for any path of length 2, there exists a chord or another path of length 2 between its two endpoints. We show that such graphs are 2-vertex-connected and moreover in the case of an edge or a vertex deletion, the distance between any pair of nonadjacent vertices remains unchanged. We give properties of hypochordal graphs, then we study the class of minimum hypochordal graphs and finally we give some complexity results for classical combinatorial problems. Keywords: graph, tree, vertex-connected, N P-complete