Hamiltonicity in Convex Bipartite Graphs release_xsbfav6tpna3no622lq7wfomkm

by P. Kowsika, V. Divya, N. Sadagopan

Released as a article .

2018  

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 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,...,x_n), is a bipartite graph such that for each y ∈ 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 monotone and 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 keil,esha, it is incorrectly claimed that Hamiltonian path problem in convex bipartite graphs is polynomial-time solvable by referring to muller which actually discusses Hamiltonian cycle (b) the algorithm appeared in 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.
In text/plain format

Archived Files and Locations

application/pdf  411.6 kB
file_nlwxhzokynhjra7ys4umildsd4
arxiv.org (repository)
web.archive.org (webarchive)
Read Archived PDF
Preserved and Accessible
Type  article
Stage   submitted
Date   2018-09-17
Version   v1
Language   en ?
arXiv  1809.06113v1
Work Entity
access all versions, variants, and formats of this works (eg, pre-prints)
Catalog Record
Revision: b59aa9ff-452b-4b94-8acd-c91420f53152
API URL: JSON