In this paper, we first present an O ( n + m ) -time sequential algorithm to solve the Hamiltonian problem on a distance-hereditary graph G, where n and m are the number of vertices and edges of G, respectively. This algorithm is faster than the previous best known algorithm for the problem which takes O ( n 2 ) time. We also give an efficient parallel implementation of our sequential algorithm. Moreover, if G is represented by its decomposition tree form, the problem can be solved optimally in O ( log n ) time using O ( ( n + m ) / log n ) processors on an EREW PRAM.