From Modular Decomposition Trees to Rooted Median Graphs
Carmen Bruckmann, Peter F. Stadler, Marc Hellmuth

TL;DR
This paper introduces a method to explain symmetric maps using rooted median graphs, extending modular decomposition trees, with applications in phylogenetics and a linear-time algorithm for construction.
Contribution
It generalizes modular decomposition trees to rooted median graphs for explaining symmetric maps and provides a linear-time algorithm for their construction.
Findings
Every symmetric map can be explained by extended hypercubes and half-grids.
A linear-time algorithm resolves prime vertices to produce explaining median graphs.
Tree-like median graphs have potential applications in phylogenetics.
Abstract
The modular decomposition of a symmetric map (or, equivalently, a set of symmetric binary relations, a 2-structure, or an edge-colored undirected graph) is a natural construction to capture key features of in labeled trees. A map is explained by a vertex-labeled rooted tree if the label coincides with the label of the last common ancestor of and in , i.e., if . Only maps whose modular decomposition does not contain prime nodes, i.e., the symbolic ultrametrics, can be exaplained in this manner. Here we consider rooted median graphs as a generalization to (modular decomposition) trees to explain symmetric maps. We first show that every symmetric map can be explained by "extended" hypercubes and half-grids. We then derive a a linear-time algorithm that stepwisely…
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