  A graph $G$ is called an $L_1$-graph if $d(u)+d(v)\ge|N(u)\cup N(v)\cup
N(w)|-1$ for every triple of vertices $u,v,w$ where $u$ and $v$ are at distance
2 and $w\in N(u)\cap N(v)$. Asratian et al. (1996) proved that all finite
connected $L_1$-graphs on at least three vertices such that $|N(u)\cap
N(v)|\ge2$ for each pair of vertices $u,v$ at distance 2 are Hamiltonian,
except for a simple family $\mathcal{K}$ of exceptions.
  We show that not all such graphs are pancyclic, but that any non-Hamiltonian
cycle in such a graph can be extended to a larger cycle containing all vertices
of the original cycle and at most two other vertices. We also prove a similar
result for paths whose endpoints do not have any common neighbors.
