## Abstract

A total dominating set of a graph G is a set D of vertices of G such that every vertex of G has a neighbor in D. A locating-total dominating set of G is a total dominating set D of G with the additional property that every two distinct vertices outside D have distinct neighbors in D; that is, for distinct vertices u and v outside D, N(u) ∩ D ≠ N(v) ∩ D where N(u) denotes the open neighborhood of u. A graph is twin-free if every two distinct vertices have distinct open and closed neighborhoods. The location-total domination number of G, denoted γ ^{L} _{t} (G), is the minimum cardinality of a locating-total dominating set in G. It is well-known that every connected graph of order n ≥ 3 has a total dominating set of size at most 2/3;n. We conjecture that if G is a twin-free graph of order n with no isolated vertex, then γ^{L} _{t} (G) ≤ 2/3n. We prove the conjecture for graphs without 4-cycles as a subgraph. We also prove that if G is a twin-free graph of order n, then γ^{L} _{t} (G) ≤ 3/4n.

Original language | English |
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Journal | Electronic Journal of Combinatorics |

Volume | 23 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2016 |

## Keywords

- Dominating sets
- Locating-dominating sets
- Total dominating sets

## ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics