# Efficient algorithms for a mixed k-partition problem of graphs without specifying bases

## Abstract

This paper describes efficient algorithms for partitioning a *k*-edge-connected graph into *k* edge-disjoint connected subgraphs, each of which has a specified number of elements(vertices and edges). If each subgraph contains the specified element (called base), we call this problem the mixed *k*-partition problem with bases(called *k*-PART-WB), otherwise we call it the mixed *k*-partition problem without bases (called *k*-PART-WOB). In this paper, we show that *k*-PART-WB always has a solution for every *k*-edge-connected graph and we consider the problem without bases and we obtain the following results: (1)for any *k*≥2, *k*-PART-WOB can be solved in *O*(∥*V*∥√∥*V*∥log_{2}∥*V*∥+∥*E*∥) time for every 4-edge-connected graph *G*=(*V,E*), (2)3-PART-WOB can be solved in *O*(∥*V*∥^{2}) for every 2-edge-connected graph *G*=(*V,E*) and (3)4-PART-WOB can be solved in *O*(∥*E*∥^{2}) for every 3-edge-connected graph *G*=(*V,E*).

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