Abstract
The extended k-partition problem is defined as follows. For the following inputs (1)an undirected graph G=(V, E)(n = ¦V¦, m= ¦E¦), (2)a vertex subset V′(⊂- V), (3)distinct vertices ai ε V′(1 ≤ i ≤ k) and (4)natural numbers n i(1 ≤ i≤k)(n1 ≤ ... ≤ n k) such that n 1 +... + n k =n′ = ¦V′¦, we compute a partition V1∪...∪V k of V and a partition V′1∪...V′ k of V′ such that (a)each V′i is included in Vi, (b)each V′i contains the specified vertex ai, (c)¦V′i¦ = ni and (d)each Vi induces a connected subgraph. If V′ = V, then the problem is called the k-partition problem. In this paper, we show that if the input graph is triconnected the extended tripartition problem can be solved in O(m + (n − n 3) · n) time and that the algorithm solves the original tripartition problem in O(m + (n 1 + n 2) · n) time. Furthermore, we show that for a k-edgeconnected graph G - (V, E) there exists a partition V1 ∪ ... V k of V such that each Vi contains the specified vertex ai, ¦Vi¦ = ni and k subgraphs G1,..., Gk are mutually edge disjoint and each of Gi contains all of elements in Vi(1 ≤ i ≤ k) and the case in which k = 3 can be solved in O(n 2) time.
Partially supported by the Grant-in-Aid for Scientific Research of the Ministry of Education, Science and Culture of Japan under Grant: (C)05680271.
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© 1994 Springer-Verlag Berlin Heidelberg
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Wada, K., Kawaguchi, K. (1994). Efficient algorithms for tripartitioning triconnected graphs and 3-edge-connected graphs. In: van Leeuwen, J. (eds) Graph-Theoretic Concepts in Computer Science. WG 1993. Lecture Notes in Computer Science, vol 790. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57899-4_47
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DOI: https://doi.org/10.1007/3-540-57899-4_47
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