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<p><b>Abstract</b>—In the literature, there are quite a few sequential and parallel algorithms to solve problems on distance-hereditary graphs. Two well-known classes of graphs, which contain trees and cographs, belong to distance-hereditary graphs. In this paper, we consider the vertex-coloring problem on distance-hereditary graphs. Let <tmath>T_{d}(|V|,|E|)</tmath> and <tmath>P_{d}(|V|,|E|)</tmath> denote the time and processor complexities, respectively, required to construct a decomposition tree representation of a distance-hereditary graph <tmath>G=(V,E)</tmath> on a PRAM model <tmath>M_{d}</tmath>. Our algorithm runs in <tmath>O(T_{d}(|V|,|E|)+\log |V|)</tmath> time using <tmath>O(P_{d}(|V|,|E|)+|V|/\log |V|)</tmath> processors on <tmath>M_{d}</tmath>. The best known result for constructing a decomposition tree needs <tmath>O(\log^{2}|V|)</tmath> time using <tmath>O(|V|+|E|)</tmath> processors on a CREW PRAM. If a decomposition tree is provided as input, we solve the problem in <tmath>O(\log |V|)</tmath> time using <tmath>O(|V|/\log |V|)</tmath> processors on an EREW PRAM. To the best of our knowledge, there is no parallel algorithm for this problem on distance-hereditary graphs.</p>
Algorithms, distance-hereditary graphs, the vertex-coloring problem, PRAM.
Sun-Yuan Hsieh, "A Simple and Fast Parallel Coloring Algorithm for Distance-Hereditary Graphs", IEEE Transactions on Parallel & Distributed Systems, vol. 14, no. , pp. 1201-1208, December 2003, doi:10.1109/TPDS.2003.1255633
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