Havel-Hakimi Theorem: A sequence s: $d_1, d_2, \ldots, d_n$ of non-negative integers with $\Delta = d_1 \geq d_2 \geq \ldots \geq d_n$ and $\Delta \geq 1$, is graphical if and only if the sequence $s_1: d_2 - 1, d_3 - 1, \ldots d_{\Delta + 1} - 1, d_{\Delta + 2}, \ldots, d_n$ is graphical.
Havel-Hakimi theorem provides an algorithm for determining whether a given finite sequence of non-negative integers is graphical. If, upon repeated application of Theorem 1, we arrive at a sequence, where every term of which 0, then the original sequence is graphical. On the other hand, if we arrive a sequence containing a negative integer, then the given sequence is not graphical.
I tried several sequences and realized, the sequence is not graphical if it fails Havel-Hakimi's theorem. However, it doesn't always work for connected graph. For instance, the sequence: $ 3, 3, 1, 1, 1, 1, 1, 1$ can be processed by Havel-Hakimi's algorithm as follows:
3, 3, 1, 1, 1, 1, 1, 1 2, 0, 0, 1, 1, 1, 1 2, 1, 1, 1, 1, 0, 0 0, 0, 1, 1, 0, 0 1, 1, 0, 0, 0, 0 0, 0, 0, 0, 0
But it can't be graphed as a connected component. On the other hand, the sequence: $5, 4, 3, 2, 2, 2, 2, 2$ also satisfies the Havel-Hakimi's algorithm, but can be graphed as follows:
So my question is, what other conditions need to be added so that Havel-Hakimi's algorithm work for connected graph? Thank you.