Pairwise compatibility graph

In graph theory, a graph $$G$$ is a pairwise compatibility graph (PCG) if there exists a tree $$T$$ and two non-negative real numbers $$d_{min} < d_{max}$$ such that each node $$u'$$ of $$G$$ has a one-to-one mapping with a leaf node $$u$$ of $$T$$ such that two nodes $$u'$$ and $$v'$$ are adjacent in $$G$$ if and only if the distance between $$u$$ and $$v$$ are in the interval $$[d_{min}, d_{max}]$$.

The subclasses of PCG include graphs of at most seven vertices, cycles, forests, complete graphs, interval graphs and ladder graphs. However, there is a graph with eight vertices that is known not to be a PCG.

Relationship to phylogenetics
Pairwise compatibility graphs were first introduced by Paul Kearney, J. Ian Munro and Derek Phillips in the context of phylogeny reconstruction. When sampling from a phylogenetic tree, the task of finding nodes whose path distance lies between given lengths $$d_{min} < d_{max}$$ is equivalent to finding a clique in the associated PCG.

Complexity
The computational complexity of recognizing a graph as a PCG is unknown as of 2020. However, the related problem of finding for a graph $$G$$ and a selection of non-edge relations $$S$$ a PCG containing $$G$$ as a subgraph and with none of the edges in $$S$$ is known to be NP-hard.

The task of finding nodes in a tree whose paths distance lies between $$d_{min}$$ and $$d_{max}$$ is known to be solvable in polynomial time. Therefore, if the tree could be recovered from a PCG in polynomial time, then the clique problem on PCGs would be polynomial too. As of 2020, neither of these complexities is known.