Exact-$2$-Relation Graphs

TL;DR

This paper characterizes and efficiently constructs a class of graphs called exact-2-relation graphs, which model evolutionary events, by linking them to block graphs and providing a tree-based representation.

Contribution

It provides a complete characterization of exact-2-relation graphs as those whose quotient is a block graph and offers an efficient method to construct their tree representations.

Findings

A graph has an exact-2-relation representation if and only if its quotient is a block graph.

An efficient algorithm exists to construct the representing tree for such graphs.

The study extends to an oriented version of these graphs.

Abstract

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.