Scattering number and modular decomposition

The scattering number of a graph G equals max{c(G\S) - \S\ S is a cutset of G} where c(G\S) denotes the number of connected components in G\S. Jung (1978) has given for any graph having no induced path on four vertices (P-4-free graph) a correspondence between the value of its scattering number and the existence of Hamiltonian paths or Hamiltonian cycles, Hochstattler and Tinhofer (to appear) studied the Hamiltonicity of P-4-sparse graphs introduced by Hoang (1985). In this paper, using modular decomposition, we show that the results of Jung and Hochstattler and Tinhofer can be generalized to a subclass of the family of semi-P-4-sparse graphs introduced in Fouquet and Giakoumakis (to appear).