Let G be a graph on nβ₯3 vertices. A graph G is almost
distance-hereditary if each connected induced subgraph H of G has the
property dHβ(x,y)β€dGβ(x,y)+1 for any pair of vertices x,yβV(H).
A graph G is called 1-heavy (2-heavy) if at least one (two) of the end
vertices of each induced subgraph of G isomorphic to K1,3β (a claw) has
(have) degree at least n/2, and called claw-heavy if each claw of G has a
pair of end vertices with degree sum at least n. Thus every 2-heavy graph is
claw-heavy. In this paper we prove the following two results: (1) Every
2-connected, claw-heavy and almost distance-hereditary graph is Hamiltonian.
(2) Every 3-connected, 1-heavy and almost distance-hereditary graph is
Hamiltonian. In particular, the first result improves a previous theorem of
Feng and Guo. Both results are sharp in some sense.Comment: 14 pages; 1 figure; a new theorem is adde