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