Parallel algorithms for Hamiltonian problems on quasi-threshold graphs

doi:10.1016/j.jpdc.2003.08.004

Journal of Parallel and Distributed Computing

Volume 64, Issue 1, January 2004, Pages 48-67

https://doi.org/10.1016/j.jpdc.2003.08.004 Get rights and content

Abstract

In this paper we show structural and algorithmic properties on the class of quasi-threshold graphs, or QT-graphs for short, and prove necessary and sufficient conditions for a QT-graph to be Hamiltonian. Based on these properties and conditions, we construct an efficient parallel algorithm for finding a Hamiltonian cycle in a QT-graph; for an input graph on n vertices and m edges, our algorithm takes $O(log n)$ time and requires O(n+m) processors on the CREW PRAM model. In addition, we show that the problem of recognizing whether a QT-graph is a Hamiltonian graph and the problem of computing the Hamiltonian completion number of a nonHamiltonian QT-graph can also be solved in $O(log n)$ time with O(n+m) processors. Our algorithms rely on $O(log n)$ -time parallel algorithms, which we develop here, for constructing tree representations of a QT-graph; we show that a QT-graph G has a unique tree representation, that is, a tree structure which meets the structural properties of G. We also present parallel algorithms for other optimization problems on QT-graphs which run in $O(log n)$ time using a linear number of processors.

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Extending previous NP-completeness results for the harmonious coloring problem and the pair-complete coloring problem on trees, bipartite graphs and cographs, we prove that these problems are also NP-complete on connected bipartite permutation graphs. We also study the $k$ -path partition problem and, motivated by a recent work of Steiner [G. Steiner, On the $k$ -path partition of graphs, Theoret. Comput. Sci. 290 (2003) 2147–2155], where he left the problem open for the class of convex graphs, we prove that the $k$ -path partition problem is NP-complete on convex graphs. Moreover, we study the complexity of these problems on two well-known subclasses of chordal graphs namely quasi-threshold and threshold graphs. Based on the work of Bodlaender [H.L. Bodlaender, Achromatic number is NP-complete for cographs and interval graphs, Inform. Process. Lett. 31 (1989) 135–138], we show NP-completeness results for the pair-complete coloring and harmonious coloring problems on quasi-threshold graphs. Concerning the $k$ -path partition problem, we prove that it is also NP-complete on this class of graphs. It is known that both the harmonious coloring problem and the $k$ -path partition problem are polynomially solvable on threshold graphs. We show that the pair-complete coloring problem is also polynomially solvable on threshold graphs by describing a linear-time algorithm.
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Parallel algorithms for Hamiltonian problems on quasi-threshold graphs

Abstract

J. Algorithms

Discrete Appl. Math.

Discrete Appl. Math.

Discrete Math.

J. Algorithms

J. Parallel Distrib. Comput.

J. Parallel Distrib. Comput.

Computers Math. Appl.

Theoret. Comput. Sci.

Inform. Process. Lett.

Inform. Process. Lett.

Discrete Math.

Parallel Computation: Models and Methods

The edge Hamiltonian problem is NP-hard

Inform. Process. Lett.

The pathwidth and treewidth of cographs

SIAM J. Discrete Math.