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**Complementarily domatic number of a graph.**
*(English)*
Zbl 0641.05045

Let D be a subset of the vertex set V(G) of a graph G. The set D is called complementarily domination if for each vertex \(x\in V(G)-D\) there exist vertices \(y\in D\), \(z\in D\) such that y is adjacent and z is non- adjacent to x in G. A complementarily domatic partition of G is a partition of V(G), all of whose classes are complementarily dominating sets in G. The maximum number of classes of a complementarily domatic partition of G is called the complementarily domatic number of G and denoted by \(d_{cp}(G)\). Thus \(d_{cp}(G)\) is a variant of the domatic number of a graph introduced by E. J. Cockayne and S. T. Hedetiemi [Networks 7, 247-261 (1977; Zbl 0384.05051)]. Here bounds for \(d_{cp}(G)\) are given for graphs G in various classes.

Reviewer: R.C.Entringer

### MSC:

05C70 | Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) |

### Keywords:

complementarily domatic partition; complementarily dominating sets; complementarily domatic number### Citations:

Zbl 0384.05051
Full Text:
EuDML

### References:

[1] | COCKAYNE E. J., HEDETNIEMI S. T.: Towards a theory of domination in graphs. Networks 7, 1977, 247-261. · Zbl 0384.05051 |

[2] | COCKAYNE E. J., DAWES R., HEDETNIEMI S. T.: Total domination in graphs. Networks 10, 1980, 211-215. · Zbl 0447.05039 |

[3] | LASKAR R., HEDETNIEMI S. T.: Connected domination in graphs. Report 414, Clemson Univ., Clemson, South Carolina, March 1983. · Zbl 0548.05055 |

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