Detour dominating set

Detour dominating set --- обходное доминирующее множество.

For a vertex [math]\displaystyle{ v }[/math] in [math]\displaystyle{ G }[/math], define

[math]\displaystyle{ D^{-}(v) = \min \{D(u,v): u \in V(G) - \{v\}\}. }[/math]

A vertex [math]\displaystyle{ u }[/math] ([math]\displaystyle{ \neq v }[/math]) is called a detour neighbor of [math]\displaystyle{ v }[/math] if [math]\displaystyle{ D(u,v) = D^{-}(v) }[/math]. The detour neighborhood [math]\displaystyle{ N_{D}(v) }[/math] of a vertex [math]\displaystyle{ v }[/math] is the set of detour neighbors of [math]\displaystyle{ v }[/math], and its closed detour neighborhood is [math]\displaystyle{ N_{D}[v] = N_{D}(v) \cup \{v\} }[/math]. A vertex [math]\displaystyle{ v }[/math] is said to detour-dominate a vertex [math]\displaystyle{ u }[/math] if [math]\displaystyle{ u = v }[/math] or [math]\displaystyle{ u }[/math] is a detour neighbor of [math]\displaystyle{ v }[/math].