A graph $G=(V,E)$ with a vertex set $V$ and an edge set $E$ is called a pairwise compatibility graph (PCG, for short) if there are a tree $T$ whose leaf set is $V$,
a non-negative edge weight $w$ in $T$, and two non-negative reals $d_{\min}\leq d_{\max}$
such that $G$ has an edge $uv\in E$ if and only if
the distance between $u$ and $v$ in the weighted tree $(T,w)$ is
in the interval $[d_{\min}, d_{\max}]$.
PCG is a new graph class motivated from bioinformatics.
In this paper, we give some
necessary and sufficient conditions for PCG based on cut-vertices and twins,
which provide reductions among PCGs.