Efficient Algorithms for the Hamiltonian Problem on Distance-Hereditary Graphs

In this paper, we first present an O(|V| + |E|)-time sequential algorithm to solve the Hamiltonian problem on a distance-hereditary graph G = (V, E). This algorithm is faster than the previous best result which takes O(|V|2) time. Let Td(|V|, |E|) and Pd(|V|, |E|) denote the parallel time and processor complexities, respectively, required to construct a decomposition tree of a distance-hereditary graph on a PRAM model Md. We also show that this problem can be solved in O(Td(|V|, |E|) + log |V|) time using O(Pd(|V|, |E|) + (|V| + |E|)/ log |V|) processors on Md. Moreover, if G is represented by its decomposition tree form, the problem can be solved optimally in O(log |V|) time using O((|V| + |E|)/ log |V|) processors on an EREW PRAM.

[1]  Bruno Courcelle,et al.  Linear Time Solvable Optimization Problems on Graphs of Bounded Clique Width , 1998, WG.

[2]  David G. Kirkpatrick,et al.  A Simple Parallel Tree Contraction Algorithm , 1989, J. Algorithms.

[3]  Gen-Huey Chen,et al.  Dynamic Programming on Distance-Hereditary Graphs , 1997, ISAAC.

[4]  Udi Rotics,et al.  On the Clique-Width of Perfect Graph Classes , 1999, WG.

[5]  M. Golumbic.  Algorithmic graph theory and perfect graphs , 1980 .

[6]  Sun-Yuan Hsieh.  Parallel Decomposition of Distance-Hereditary Graphs , 1999, ACPC.

[7]  Maw-Shang Chang,et al.  Hamiltonian Cycle Problem on Distance-Hereditary Graphs , 2003, J. Inf. Sci. Eng..

[8]  A. Brandstädt,et al.  A linear-time algorithm for connected r-domination and Steiner tree on distance-hereditary graphs , 1998 .

[9]  Tsan-sheng Hsu,et al.  Characterization of Efficiently Solvable Problems on Distance-Hereditary Graphs , 1998, ISAAC.

[10]  Peter L. Hammer,et al.  Completely separable graphs , 1990, Discret. Appl. Math..

[11]  Marina Moscarini,et al.  Distance-Hereditary Graphs, Steiner Trees, and Connected Domination , 1988, SIAM J. Comput..

[12]  Haiko Müller,et al.  Polynomial Time Algorithms for Hamiltonian Problems on Bipartite Distance-Hereditary Graphs , 1993, Inf. Process. Lett..

[13]  Tsan-sheng Hsu,et al.  A Faster Implementation of a Parallel Tree Contraction Scheme and Its Application on Distance-Hereditary Graphs , 2000, J. Algorithms.

[14]  Tsan-sheng Hsu,et al.  Efficient Parallel Algorithms on Distance Hereditary Graphs , 1999, Parallel Process. Lett..

[15]  Hans-Jürgen Bandelt,et al.  Distance-hereditary graphs , 1986, J. Comb. Theory B.

[16]  Tsan-sheng Hsu,et al.  Efficient parallel algorithms on distance-hereditary graphs , 1997, Proceedings of the 1997 International Conference on Parallel Processing (Cat. No.97TB100162).

[17]  E. Howorka.  A CHARACTERIZATION OF DISTANCE-HEREDITARY GRAPHS , 1977 .

[18]  Feodor F. Dragan,et al.  Dominating Cliques in Distance-Hereditary Graphs , 1994, SWAT.