
Florent Foucaud

Michael A. Henning
Keywords:
Locatingdominating sets, Total dominating sets, Dominating sets
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 locatingtotal 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 $D$, $N(u) \cap D \ne N(v) \cap D$ where $N(u)$ denotes the open neighborhood of $u$. A graph is twinfree if every two distinct vertices have distinct open and closed neighborhoods. The locationtotal domination number of $G$, denoted $\gamma_t^L(G)$, is the minimum cardinality of a locatingtotal dominating set in $G$. It is wellknown that every connected graph of order $n \ge 3$ has a total dominating set of size at most $\frac{2}{3}n$. We conjecture that if $G$ is a twinfree graph of order $n$ with no isolated vertex, then $\gamma_t^L(G) \le \frac{2}{3}n$. We prove the conjecture for graphs without $4$cycles as a subgraph. We also prove that if $G$ is a twinfree graph of order $n$, then $\gamma_t^L(G) \le \frac{3}{4}n$.