Hamiltonian Paths in the Square of a Tree

We introduce a new family of graphs for which the Hamiltonian path problem is non-trivial and yet has a linear time solution. The square of a graph G = (V, E), denoted as G, is a graph with the set of vertices V , in which two vertices are connected by an edge if there exists a path of length at most 2 connecting them in G. Harary & Schwenk (1971) proved that the square of a tree T contains a Hamiltonian cycle if and only if T is a caterpillar, i.e., it is a single path with several leafs connected to it. Our first main result is a simple graph-theoretic characterization of trees T for which T 2 contains a Hamiltonian path: T 2 has a Hamiltonian path if and only if T is a horsetail (the name is due to the characteristic shape of these trees, see Figure 1). Our next results are two efficient algorithms: linear time testing if T 2 contains a Hamiltonian path and finding such a path (if there is any), and linear time preprocessing after which we can check for any pair (u, v) of nodes of T in constant time if there is a Hamiltonian path from u to v in T .