Let $G$ be a graph on $n\geq 3$ vertices. A graph $G$ is almost
distance-hereditary if each connected induced subgraph $H$ of $G$ has the
property $d_{H}(x,y)\leq d_{G}(x,y)+1$ for any pair of vertices $x,y\in V(H)$.
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 $K_{1,3}$ (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