  Pairwise compatibility graphs (PCGs) with non-negative integer edge weights
recently have been used to describe rare evolutionary events and scenarios with
horizontal gene transfer. Here we consider the case that vertices are separated
by exactly two discrete events: Given a tree $T$ with leaf set $L$ and
edge-weights $\lambda: E(T)\to\mathbb{N}_0$, the non-negative integer pairwise
compatibility graph $\textrm{nniPCG}(T,\lambda,2,2)$ has vertex set $L$ and
$xy$ is an edge whenever the sum of the non-negative integer weights along the
unique path from $x$ to $y$ in $T$ equals $2$. A graph $G$ has a representation
as $\textrm{nniPCG}(T,\lambda,2,2)$ if and only if its point-determining
quotient $G/\!\rthin$ is a block graph, where two vertices are in relation
$\rthin$ if they have the same neighborhood in $G$. If $G$ is of this type, a
labeled tree $(T,\lambda)$ explaining $G$ can be constructed efficiently. In
addition, we consider an oriented version of this class of graphs.
