On the Galois Lattice of Bipartite Distance Hereditary Graphs

TL;DR

This paper characterizes bipartite graphs with tree-like Galois lattices, focusing on Bipartite Distance Hereditary graphs, and introduces a compact encoding enabling efficient computations of graph properties.

Contribution

It provides a complete characterization of bipartite graphs with tree-like Galois lattices and offers a new compact encoding for Bipartite Distance Hereditary graphs.

Findings

Galois lattice of bipartite distance hereditary graphs is tree-like after removing extremal elements

The lattice can be represented via containment among directed paths in an arborescence

Proposed encoding enables optimal time computation of neighborhood intersections and bicliques

Abstract

We give a complete characterization of bipartite graphs having tree-like Galois lattices. We prove that the poset obtained by deleting bottom and top elements from the Galois lattice of a bipartite graph is tree-like if and only if the graph is a Bipartite Distance Hereditary graph. By relying on the interplay between bipartite distance hereditary graphs and series-parallel graphs, we show that the lattice can be realized as the containment relation among directed paths in an arborescence. Moreover, a compact encoding of Bipartite Distance Hereditary graphs is proposed, that allows optimal time computation of neighborhood intersections and maximal bicliques.