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IEICE Transactions on Information and Systems 2008 E91-D(2):170-177; doi:10.1093/ietisy/e91-d.2.170
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Copyright © 2008 The Institute of Electronics, Information and Communication Engineers

Special Section on Foundations of Computer Science -- Papers -- Graph Algorithms

Longest Path Problems on Ptolemaic Graphs*

Yoshihiro TAKAHARA1,2, Sachio TERAMOTO1,3 and Ryuhei UEHARA1

1 The authors are with School of Information Science, JAIST, Nomi-shi, 923–1292 Japan. E-mail: uehara{at}jaist.ac.jp, 2 Presently, with Sekisui House., 3 Presently, with Service Platforms Research Laboratories, NEC Corporation.


  Abstract

Longest path problem is a problem for finding a longest path in a given graph. While the graph classes in which the Hamiltonian path problem can be solved efficiently are widely investigated, there are few known graph classes such that the longest path problem can be solved efficiently. Polynomial time algorithms for finding a longest cycle and a longest path in a Ptolemaic graph are proposed. Ptolemaic graphs are the graphs that satisfy the Ptolemy inequality, and they are the intersection of chordal graphs and distance-hereditary graphs. The algorithms use the dynamic programming technique on a laminar structure of cliques, which is a recent characterization of Ptolemaic graphs.

Key Words: dynamic programming, Hamiltonian path/cycle problem, longest path/cycle problem, Ptolemaic graphs

Manuscript received March 27, 2007. Manuscript revised June 18, 2007.

* An extended abstract was presented at KyotoCGGT 2007. The authors are partially supported by the Ministry, Grant-in-Aid for Scientific Research (C).


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