Locating-Total Dominating Sets in Twin-Free Graphs: a Conjecture

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 6= 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 γL 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 3n. We conjecture that if G is a twin-free graph of order n with no isolated vertex, then γL t (G) 6 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 γL t (G) 6 3 4n.