Pairwise compatibility graph

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Graph (b) that is pairwise compatibility graphs of the trees (a) and (c)

Graphs that are not pairwise compatibility graphs

In graph theory, a graph ${\displaystyle G}$ is a pairwise compatibility graph (PCG) if there exists a tree ${\displaystyle T}$ and two non-negative real numbers ${\displaystyle d_{min}<d_{max}}$ such that each node ${\displaystyle u'}$ of ${\displaystyle G}$ has a one-to-one mapping with a leaf node ${\displaystyle u}$ of ${\displaystyle T}$ such that two nodes ${\displaystyle u'}$ and ${\displaystyle v'}$ are adjacent in ${\displaystyle G}$ if and only if the distance between ${\displaystyle u}$ and ${\displaystyle v}$ are in the interval ${\displaystyle [d_{min},d_{max}]}$.^[1]

The subclasses of PCG include graphs of at most seven vertices, cycles, forests, complete graphs, interval graphs and ladder graphs.^[1] However, there is a graph with eight vertices that is known not to be a PCG.^[2]

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 ${\displaystyle d_{min}<d_{max}}$ is equivalent to finding a clique in the associated PCG.^[3]

The computational complexity of recognizing a graph as a PCG is unknown as of 2020.^[1] However, the related problem of finding for a graph ${\displaystyle G}$ and a selection of non-edge relations ${\displaystyle S}$ a PCG containing ${\displaystyle G}$ as a subgraph and with none of the edges in ${\displaystyle S}$ is known to be NP-hard.^[2]

The task of finding nodes in a tree whose paths distance lies between ${\displaystyle d_{min}}$ and ${\displaystyle d_{max}}$ 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.^[1]