Admissible sequence

Admissible sequence — допустимая последовательность.

Let [math]\displaystyle{ \,G = (V,E) }[/math] be a simple undirected graph of order [math]\displaystyle{ \,n }[/math]. Let [math]\displaystyle{ \tau - (n_{1}, \ldots, n_{k}) }[/math] denote a sequence of positive integers such that [math]\displaystyle{ n_{1} + \ldots + n_{k} = n }[/math]. Such a sequence will be called admissible for [math]\displaystyle{ \,G }[/math]. If [math]\displaystyle{ \tau = (n_{1}, \ldots, n_{K}) }[/math] is an admissible sequence for [math]\displaystyle{ \,G }[/math] and there exists a partition [math]\displaystyle{ (V_{1}, \ldots, V_{k}) }[/math] of the vertex set [math]\displaystyle{ \,V }[/math] such that for each [math]\displaystyle{ i \in \{1, \ldots, k\} }[/math], [math]\displaystyle{ \,|V_{i}| = n_{i} }[/math] and the subgraph induced by [math]\displaystyle{ \,V_{i} }[/math] is connected, then [math]\displaystyle{ \,\tau }[/math] is called realizable in [math]\displaystyle{ \,G }[/math] and the sequence [math]\displaystyle{ (V_{1}, \ldots, V_{k}) }[/math] is said to be [math]\displaystyle{ \,G }[/math]-realization of [math]\displaystyle{ \,\tau }[/math] or a realization of [math]\displaystyle{ \,\tau }[/math] in [math]\displaystyle{ \,G }[/math]. A graph [math]\displaystyle{ \,G }[/math] is arbitrarily vertex decomposable if for each admissible sequence [math]\displaystyle{ \,\tau }[/math] for [math]\displaystyle{ \,G }[/math] there exists a [math]\displaystyle{ \,G }[/math]-realization of [math]\displaystyle{ \,\tau }[/math].