Abstract
We show that anyk-connected graphG = (V, E) has a sparsek-connected spanning subgraphG′ = (V, E′) with ¦E′¦ =O(k¦V¦) by presenting anO(¦E¦)-time algorithm to find one such subgraph, where connectivity stands for either edge-connectivity or node-connectivity. By using this algorithm as preprocessing, the time complexities of some graph problems related to connectivity can be improved. For example, the current best time boundO(max{k 2¦V¦1/2,k¦V¦}¦E¦) to determine whether node-connectivityK(G) of a graphG = (V, E) is larger than a given integerk or not can be reduced toO(max{k 3¦V¦3/2,k 2¦V¦2}).
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Communicated by Greg N. Frederickson.
The first author was partially supported by the Grant-in-Aid for Encouragement of Young Scientists of the Ministry of Education, Science and Culture of Japan and by the subvention to young scientists by the Research Foundation of Electrotechnology of Chubu.
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Nagamochi, H., Ibaraki, T. A linear-time algorithm for finding a sparsek-connected spanning subgraph of ak-connected graph. Algorithmica 7, 583–596 (1992). https://doi.org/10.1007/BF01758778
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DOI: https://doi.org/10.1007/BF01758778