Discussiones Mathematicae Graph Theory 33(2) (2013)
347-359

doi: 10.7151/dmgt.1666

Shinya Fujita
Department of Integrated Design Engineering | Henry Liu
Departamento de Matemática |

**Keywords:** graph decomposition, vertex colouring, *k*-connected

**2010 Mathematics Subject Classification:** 05C15, 05C40, 05C70.

[1] | B. Bollobás, Modern Graph Theory (Springer-Verlag, New York, 1998). |

[2] | R.J. Duffin, Topology of series-parallel networks, J. Math. Anal. Appl. 10 (1965) 303--318. |

[3] | E.S. Elmallah and C.J. Colbourn, Series-parallel subgraphs of planar graphs, Networks 22 (1992) 607--614, doi: 10.1002/net.3230220608. |

[4] | S. Fujita and H. Liu, The balanced decomposition number and vertex connectivity, SIAM. J. Discrete Math. 24 (2010) 1597--1616, doi: 10.1137/090780894. |

[5] | S. Fujita and H. Liu, Further results on the balanced decomposition number, in: Proceedings of the Forty-First Southeastern International Conference on Combinatorics, Graph Theory and Computing, Congr. Numer. 202 (2010) 119--128. |

[6] | S. Fujita and T. Nakamigawa, Balanced decomposition of a vertex-coloured graph, Discrete Appl. Math. 156 (2008) 3339--3344, doi: 10.1016/j.dam.2008.01.006. |

Received 24 September 2011

Revised 13 April 2012

Accepted 16 April 2012