On $P_5$-free Chordal bipartite graphs

TL;DR

This paper studies $P_5$-free chordal bipartite graphs, revealing their structure and providing polynomial-time algorithms for classical problems like Hamiltonian cycle, longest path, Steiner path, and minimum leaf spanning tree.

Contribution

It introduces the Nested Neighborhood Ordering for $P_5$-free chordal bipartite graphs and develops efficient algorithms for several classical problems within this class.

Findings

Existence of Nested Neighborhood Ordering in these graphs

Polynomial-time algorithms for Hamiltonian cycle and path

Polynomial algorithms for Steiner path and minimum leaf spanning tree

Abstract

A bipartite graph is chordal bipartite if every cycle of length at least 6 has a chord in it. In this paper, we investigate the structure of $P_5$-free chordal bipartite graphs and show that these graphs have a Nested Neighborhood Ordering, a special ordering among its vertices. Further, using this ordering, we present polynomial-time algorithms for classical problems such as Hamiltonian cycle (path) and longest path. Two variants of Hamiltonian path include Steiner path and minimum leaf spanning tree, and we obtain polynomial-time algorithms for these problems as well restricted to $P_5$-free chordal bipartite graphs.