Fault-tolerant meshes with constant degree

This paper proves that for every positive integers n, k and any positive constant E, we can explicitly construct a graph G with n + 0(k(1+epsilon)) vertices and a constant degree such that even after removing any k vertices from G, the remaining graph still contains an n-vertex dipath. This paper also proves that for every positive integers n, k and any positive constant epsilon, we can explicitly construct a graph H with n + O(k(2+epsilon)) vertices and a constant degree such that even after removing any k vertices from H, the remaining graph still contains an n-vertex 2-dimensional square mesh.