The longest path problem is a generalization of the well known Hamiltonian path problem. It has been shown that the longest path problem is NP-complete. Many researchers try to solve it on some special classes of graphs. Another practical generalization is given two vertices and tries to find a longest path between these two vertices. It is called the end-to-end longest path problem. In this paper, we consider this general problem on a mesh. We generalize a previous work to solve the problem on meshes with one missing vertex in linear time. Further research opportunities and improvements are also suggested.