## Abstract

A set S of vertices in a graph G is a total dominating set of G if every vertex is adjacent to a vertex in S. A fundamental problem in total domination theory in graphs is to determine which graphs have two disjoint total dominating sets. In this paper, we solve this problem by providing a constructive characterization of the graphs that have two disjoint total dominating sets. Our characterization gives an entirely new description of graphs with two disjoint total dominating sets and places them in another context, developing them from four base graphs and applies a sequence of operations from seventeen operations that are independent and necessary to produce all such graphs. We show that every graph with two disjoint total dominating sets can be constructed using this method.

Original language | English |
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Pages (from-to) | 359-375 |

Number of pages | 17 |

Journal | Ars Mathematica Contemporanea |

Volume | 16 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2019 |

## Keywords

- Disjoint total dominating sets
- Total domination number

## ASJC Scopus subject areas

- Theoretical Computer Science
- Algebra and Number Theory
- Geometry and Topology
- Discrete Mathematics and Combinatorics