Is this path finding problem in a 01-matrix NP-complete?
Your problem is to determine if there is a Hamiltonian path in a grid graph (an induced subgraph of a rectangular grid). This is NP-complete: see Itai, Papadimitriou and Szwarcfiter, Hamilton Paths in Grid Graphs (SIAM J. Comput., 11(4):676&ndash;686, 1982) PDF.