Certifying algorithms for the path cover and related problems on interval graphs

  • Authors:

  • Ruo-Wei Hung;Maw-Shang Chang

  • Affiliations:

  • Department of Computer Science and Information Engineering, Chaoyang University of Technology, Wufong, Taichung, Taiwan;Department of Computer Science and Information Engineering, National Chung Cheng University, Ming-Hsiung, Chiayi, Taiwan

  • Venue:
  • ICCSA'10 Proceedings of the 2010 international conference on Computational Science and Its Applications - Volume Part II
  • Year:
  • 2010

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Abstract

A certifying algorithm for a problem is an algorithm that provides a certificate with each answer that it produces. The certificate is a piece of evidence that proves the answer has no compromised by a bug in the implementation. A Hamiltonian cycle in a graph is a simple cycle in which each vertex of the graph appears exactly once. The Hamiltonian cycle problem is to test whether a graph has a Hamiltonian cycle. A path cover of a graph is a family of vertex-disjoint paths that covers all vertices of the graph. The path cover problem is to find a path cover of a graph with minimum cardinality. The scattering number of a noncomplete connected graph G=(V,E) is defined by s(G)= max {ω(G−S)−|S|: S⊆V and $\omega(G-S)\geqslant 1\}$, in which ω(G−S) denotes the number of components of the graph G−S. The scattering number problem is to determine the scattering number of a graph. A recognition problem of graphs is to decide whether a given input graph has a certain property. To the best of our knowledge, most published certifying algorithms are to solve the recognition problems for special classes of graphs. This paper presents O(n)-time certifying algorithms for the above three problems, including Hamiltonian cycle problem, path cover problem, and scattering number problem, on interval graphs given a set of n intervals with endpoints sorted. The certificates provided by our algorithms can be authenticated in O(n) time.