Bounds on the Locating-Domination Number and Differentiating-Total Domination Number in Trees

A subset S of vertices in a graph G = (V,E) is a dominating set of G if every vertex in V − S has a neighbor in S, and is a total dominating set if every vertex in V has a neighbor in S. A dominating set S is a locating-dominating set of G if every two vertices x, y ∈ V − S satisfy N(x) ∩ S ≠ N(y) ∩ S. The locating-domination number γL(G) is the minimum cardinality of a locating-dominating set of G. A total dominating set S is called a differentiating-total dominating set if for every pair of distinct vertices u and v of G, N[u] ∩ S ≠ N[v] ∩ S. The minimum cardinality of a differentiating-total dominating set of G is the differentiating-total domination number of G, denoted by γtD(G)$\gamma _t^D (G)$. We obtain new upper bounds for the locating-domination number, and the differentiating-total domination number in trees. Moreover, we characterize all trees achieving equality for the new bounds

Bounds on the Locating-Domination Number and Differentiating-Total Domination Number in Trees

A subset $S$ of vertices in a graph $G = (V,E)$ is a dominating set of $G$ if every vertex in $V − S$ has a neighbor in $S$, and is a total dominating set if every vertex in $V$ has a neighbor in $S$. A dominating set $S$ is a locating-dominating set of $G$ if every two vertices $x, y \in V − S$ satisfy $N(x) \cap S \ne N(y) \cap S$. The locating-domination number $\gamma_L (G)$ is the minimum cardinality of a locating-dominating set of $G$. A total dominating set $S$ is called a differentiating-total dominating set if for every pair of distinct vertices $u$ and $v$ of $G$, $N[u] \cap S \ne N[v] \cap S$. The minimum cardinality of a differentiating-total dominating set of $G$ is the differentiating-total domination number of $G$, denoted by $\gamma_t^D (G)$. We obtain new upper bounds for the locating-domination number, and the differentiating-total domination number in trees. Moreover, we characterize all trees achieving equality for the new bounds