On Computing the Galois Lattice of Bipartite Distance Hereditary Graphs

TL;DR

This paper presents a linear-time algorithm for computing the Galois lattice of Bipartite Distance Hereditary graphs, leveraging their structural properties to improve efficiency over previous methods.

Contribution

It provides a sharp estimate on the number of maximal bicliques in BDH graphs and introduces an $O(m)$ time algorithm for Galois lattice computation.

Findings

Galois lattice can be built in $O(m\times n)$ worst case for domino-free graphs.

BDH graphs have a linear number of maximal bicliques.

The proposed algorithm computes the Galois lattice in $O(m)$ time.

Abstract

The class of Bipartite Distance Hereditary (BDH) graphs is the intersection between bipartite domino-free and chordal bipartite graphs.\ Graphs in both the latter classes have linearly many maximal bicliques, implying the existence of polynomial-time algorithms for computing the associated Galois lattice.\ Such a lattice can indeed be built in $O(m\times n)$ worst case-time for a domino-free graph with $m$ edges and $n$ vertices.\ In this paper we give a sharp estimate on the number of the maximal bicliques of BDH graphs and exploit such result to give an $O(m)$ worst case time algorithm for computing the Galois lattice of BDH graphs. By relying on the fact that neighborhoods of vertices of BDH graphs can be realized as directed paths in a arborescence, we give an $O(n)$ worst-case space and time encoding of both the input graph and its Galois lattice, provided that the reverse of a Bandelt and Mulder building sequence is given.