Locating set

Locating set --- размещённое множество.

Let [math]\displaystyle{ S = \{v_{1}, \ldots, v_{k}\} }[/math] be a set of vertices in a connected graph [math]\displaystyle{ G }[/math] and let [math]\displaystyle{ v \in V(G) }[/math]. The [math]\displaystyle{ k }[/math]-vector (ordered [math]\displaystyle{ k }[/math]-tuple) [math]\displaystyle{ c_{S}(v) }[/math] of [math]\displaystyle{ v }[/math] with respect to [math]\displaystyle{ S }[/math] is defined by

[math]\displaystyle{ c_{S}(v) = (d(v,v_{1}), \ldots, d(v,v_{k})), }[/math]

where [math]\displaystyle{ d(v,v_{i}) }[/math] is the distance between [math]\displaystyle{ v }[/math] and [math]\displaystyle{ v_{i} }[/math] ([math]\displaystyle{ 1 \leq i \leq k }[/math]). The set [math]\displaystyle{ S }[/math] is called a locating set if the [math]\displaystyle{ k }[/math]-vectors [math]\displaystyle{ c_{S}(v) }[/math], [math]\displaystyle{ v \in V(G) }[/math], are distinct. The location number [math]\displaystyle{ loc(G) }[/math] of [math]\displaystyle{ G }[/math] is the minimum cardinality of a locating set in [math]\displaystyle{ G }[/math].