N-mesh

[math]\displaystyle{ n }[/math]-mesh --- [math]\displaystyle{ n }[/math]-сеть.

An [math]\displaystyle{ n }[/math]-dimensional mesh (abbreviated [math]\displaystyle{ n }[/math]-mesh) is the Cartesian product of [math]\displaystyle{ n }[/math] path graphs [math]\displaystyle{ P_{r_{1}}, \ldots, P_{r_{n}} }[/math] of orders [math]\displaystyle{ r_{i} }[/math] and is denoted by [math]\displaystyle{ M(r_{1}, \ldots, r_{n}) }[/math]. Thus, [math]\displaystyle{ V(M) = \{(a_{1}, \ldots, a_{n})| \; (1 \leq a_{i} \leq r_{i})\} }[/math] and for [math]\displaystyle{ x,y \in V(M) }[/math], [math]\displaystyle{ (x,y) \in E(M) }[/math] if and only if

[math]\displaystyle{ \sum_{i=1}^{m} |x_{i} - y_{i}| = 1. }[/math]

A mesh [math]\displaystyle{ M(2,b) }[/math] is called a ladder.

The boundary of a 2-mesh [math]\displaystyle{ M(a,b) }[/math] is defined as the outer cycle of [math]\displaystyle{ M(a,b) }[/math], it has length [math]\displaystyle{ 2a + 2b - 4 }[/math] and is the cycle through the vertices of degree 2 or 3 in [math]\displaystyle{ M(a,b) }[/math]. A submesh [math]\displaystyle{ M(c,d) }[/math] of the mesh [math]\displaystyle{ M(a,b) }[/math] such that [math]\displaystyle{ c = a }[/math] or [math]\displaystyle{ b = d }[/math] is called a contraction of [math]\displaystyle{ M(a,b) }[/math] and [math]\displaystyle{ M(a,b) }[/math] is said to be contracted to [math]\displaystyle{ M(c,d) }[/math].