From modular decomposition trees to rooted median graphs

• Published in: Discrete Applied Mathematics Vol. 310; pp. 1 - 9
• Main Authors: Bruckmann, Carmen, Stadler, Peter F., Hellmuth, Marc
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 31.03.2022; Elsevier BV
• Subjects: 2-structures; Algorithm; Algorithms; Apexes; Decomposition; Graph coloring; Graphs; Half-grid; Hypercube; Median graph; Modular decomposition; Prime module; Prime vertex replacement; Symbolic ultrametrics; Trees (mathematics); Prime vertex replacement; Symbolic ultrametrics; 2-structures; Hypercube; Modular decomposition; Prime module; Half-grid; Algorithm; Median graph
• ISSN: 0166-218X; 1872-6771; 1872-6771
• DOI: 10.1016/j.dam.2021.12.017

The modular decomposition of a symmetric map δ:X×X→Υ (or, equivalently, a set of pairwise-disjoint symmetric binary relations, a 2-structure, or an edge-colored undirected graph) is a natural construction to capture key features of δ in terms of a labeled tree. A map δ is explained by a vertex-labeled rooted tree (T,t) if the label δ(x,y) coincides with the label of the lowest common ancestor of x and y in T, i.e., if δ(x,y)=t(lca(x,y)). Only maps whose modular decomposition does not contain prime nodes, i.e., the symbolic ultrametrics, can be explained in this manner. Here we consider rooted median graphs as a generalization of (modular decomposition) trees to explain symmetric maps. We derive a linear-time algorithm that stepwisely resolves prime vertices in the modular decomposition tree to obtain a rooted and labeled median graph that explains a given symmetric map δ.