Hamiltonian problem on claw-free and almost distance-hereditary graphs

Let G=(V,E)G=(V,E) be a connected graph. The distance between two vertices xx and yy in GG, denoted by dG(x,y)dG(x,y), is the length of a shortest path between xx and yy. A graph GG is called almost distance-hereditary, if each connected induced subgraph HH of GG has the property that dH(u,v)≤dG(u,v)+1dH(u,v)≤dG(u,v)+1 for every pair of vertices uu and vv in HH. We will confirm that every 2-connected, claw-free and almost distance-hereditary graph has a Hamiltonian cycle.