  The Reconstruction Conjecture due to Kelly and Ulam states that every graph
with at least 3 vertices is uniquely determined by its multiset of subgraphs
$\{G-v: v\in V(G)\}$. Let $diam(G)$ and $\kappa(G)$ denote the diameter and the
connectivity of a graph $G$, respectively, and let $\mathcal{G}_2:=\{G:
\textrm{diam}(G)=2\}$ and
$\mathcal{G}_3:=\{G:\textrm{diam}(G)=\textrm{diam}(\overline{G})=3\}$. It is
known that the Reconstruction Conjecture is true if and only if it is true for
every 2-connected graph in $\mathcal{G}_2\cup \mathcal{G}_3$. Balakumar and
Monikandan showed that the Reconstruction Conjecture holds for every
triangle-free graph $G$ in $\mathcal{G}_2\cup \mathcal{G}_3$ with
$\kappa(G)=2$. Moreover, they asked whether the result still holds if
$\kappa(G)\ge 3$. (If yes, the class of graphs critical for solving the
Reconstruction Conjecture is restricted to 2-connected graphs in
$\mathcal{G}_2\cup\mathcal{G}_3$ which contain triangles.) In this paper, we
give a partial solution to their question by showing that the Reconstruction
Conjecture holds for every triangle-free graph $G$ in $\mathcal{G}_3$ and every
triangle-free graph $G$ in $\mathcal{G}_2$ with $\kappa(G)=3$. We also prove
similar results about the Edge Reconstruction Conjecture.
