Local tree-width

Local tree-width --- локальная древесная ширина.

We define the [math]\displaystyle{ r }[/math]-neighborhood [math]\displaystyle{ N_{r}(v) }[/math] of a vertex [math]\displaystyle{ v \in V(G) }[/math] to be the set of all vertices [math]\displaystyle{ w \in V(G) }[/math] of distance at most [math]\displaystyle{ r }[/math] from [math]\displaystyle{ v }[/math], and we let [math]\displaystyle{ \langle N_{r}(v)\rangle }[/math] denote the subgraph induced by [math]\displaystyle{ G }[/math] on [math]\displaystyle{ N_{r}(v) }[/math]. Then, denoting the tree-width of a graph [math]\displaystyle{ H }[/math] by [math]\displaystyle{ tw(H) }[/math], we let

[math]\displaystyle{ ltw^{G}(r) = \max\{tw(\langle N_{r}(v)\rangle) | \; v \in V(G). }[/math]

[math]\displaystyle{ ltw^{G}(r) }[/math] is called local tree-width.