Let $x$ and $y$ be two distinct vertices in a connected graph $G$. The $x,y$-location of a vertex $w$ is the ordered pair of distances from $w$ to $x$ and $y$, that is, the ordered pair $(d(x,w), d(y,w))$. A set of vertices $W$ in $G$ is $x,y$-located if any two vertices in $W$ have distinct $x,y$-location.A set $W$ of vertices in $G$ is 2-located if it is $x,y$-located, for some distinct...

Let M = {v1, v2 ... vl} be an ordered set of vertices in a graph G. Then (d(u, v1), d(u, v2) ... d(u, vl)) is called the M-location of a vertex u of G. The set M is called a locating set if the vertices of G have distinct M-locations. A minimum locating set is a set M with minimum cardinality. The cardinality of a minimum locating set of G is called Location Number L(G). This concept has wide a...

We study the problems Locating-Dominating Set and Metric Dimension, which consist in determining a minimum-size set of vertices that distinguishes the vertices of a graph using either neighbourhoods or distances. We consider these problems when restricted to interval graphs and permutation graphs. We prove that both decision problems are NP-complete, even for graphs that are at the same time in...

A dominating set S of graph G is called metric-locating-dominating if it is also locating, that is, if every vertex v is uniquely determined by its vector of distances to the vertices in S . If moreover, every vertex v not in S is also uniquely determined by the set of neighbors of v belonging to S , then it is said to be locating-dominating. Locating, metric-locating-dominating and locatingdom...

A total dominating set of a graph G = (V,E) with no isolated vertex is a set D ⊆ V (G) such that every vertex is adjacent to a vertex in D. A total dominating set D of G is a locating-total dominating set if for every pair of distinct vertices u and v in V −D, N(u) ∩D = N(v) ∩D. Let γ L(G) be the minimum cardinality of a locating-total dominating set of G. We show that for a nontrivial tree T o...