Efficient algorithms for tripartitioning triconnected graphs and 3-edge-connected graphs

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 a_i 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 V_i, (b)each V′_i contains the specified vertex a_i, (c)¦V′_i¦ = n_i and (d)each V_i 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 V_i contains the specified vertex a_i, ¦V_i¦ = n_i and k subgraphs G₁,..., G_k are mutually edge disjoint and each of G_i contains all of elements in V_i(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.