Hamiltonicity in Convex Bipartite Graphs

TL;DR

This paper characterizes when convex bipartite graphs contain Hamiltonian cycles and paths, providing linear-time algorithms for certain subclasses, and clarifies misconceptions about the problem's complexity.

Contribution

It offers a necessary and sufficient condition for Hamiltonian cycles in convex bipartite graphs and introduces linear-time algorithms for monotone subclasses.

Findings

Linear-time algorithm for Hamiltonian cycle in convex bipartite graphs.

Chvatal's condition is sufficient for convex bipartite graphs.

Hamiltonian path problem remains open in non-monotone convex bipartite graphs.

Abstract

For a connected graph, the Hamiltonian cycle (path) is a simple cycle (path) that spans all the vertices in the graph. It is known from \cite{muller,garey} that HAMILTONIAN CYCLE (PATH) are NP-complete in general graphs and chordal bipartite graphs. A convex bipartite graph $G$ with bipartition $(X,Y)$ and an ordering $X=(x_1,\ldots,x_n)$, is a bipartite graph such that for each $y \in Y$, the neighborhood of $y$ in $X$ appears consecutively. $G$ is said to have convexity with respect to $X$. Further, convex bipartite graphs are a subclass of chordal bipartite graphs. In this paper, we present a necessary and sufficient condition for the existence of a Hamiltonian cycle in convex bipartite graphs and further we obtain a linear-time algorithm for this graph class. We also show that Chvatal's necessary condition is sufficient for convex bipartite graphs. The closely related problem is HAMILTONIAN PATH whose complexity is open in convex bipartite graphs. We classify the class of convex bipartite graphs as {\em monotone} and {\em non-monotone} graphs. For monotone convex bipartite graphs, we present a linear-time algorithm to output a Hamiltonian path. We believe that these results can be used to obtain algorithms for Hamiltonian path problem in non-monotone convex bipartite graphs. It is important to highlight (a) in \cite{keil,esha}, it is incorrectly claimed that Hamiltonian path problem in convex bipartite graphs is polynomial-time solvable by referring to \cite{muller} which actually discusses Hamiltonian cycle (b) the algorithm appeared in \cite{esha} for the longest path problem (Hamiltonian path problem) in biconvex and convex bipartite graphs have an error and it does not compute an optimum solution always. We present an infinite set of counterexamples in support of our claim.