Hamiltonicity: Variants and Generalization in P_5-free Chordal Bipartite graphs

07/10/2021 ∙ by S Aadhavan, et al. ∙ 0

A bipartite graph is chordal bipartite if every cycle of length at least six has a chord in it. Müller <cit.> has shown that the Hamiltonian cycle problem is NP-complete on chordal bipartite graphs by presenting a polynomial-time reduction from the satisfiability problem. The microscopic view of the reduction instances reveals that the instances are P_9-free chordal bipartite graphs, and hence the status of Hamiltonicity in P_8-free chordal bipartite graphs is open. In this paper, we identify the first non-trivial subclass of P_8-free chordal bipartite graphs which is P_5-free chordal bipartite graphs, and present structural and algorithmic results on P_5-free chordal bipartite graphs. We investigate the structure of P_5-free chordal bipartite graphs and show that these graphs have a Nested Neighborhood Ordering (NNO), a special ordering among its vertices. Further, using this ordering, we present polynomial-time algorithms for classical problems such as the Hamiltonian cycle (path), also the variants and generalizations of the Hamiltonian cycle (path) problem. We also obtain polynomial-time algorithms for treewidth (pathwidth), and minimum fill-in in P_5-free chordal bipartite graph. We also present some results on complement graphs of P_5-free chordal bipartite graphs.

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