A sequence of integers $d_1, \dots, d_n$ is called graphical if there exists a simple graph $G$ with it as its degree sequence. Deciding if a sequence is graphical is called the Graph Realization Problem.
A theorem by Havel and Hakimi gives an algorithm to construct such a graph if it exists: it proceeds by repeatedly selecting a vertex of highest degree $v$ and connecting it to vertices of high degree until the degree of $v$ is depleted.
I want instead to consider the algorithm that, at every step, connects a vertex of highest degree with a single vertex of lowest nonzero degree. Note that this algorithm, unlike the Havel-Hakimi one, does not select a vertex of the highest degree and connects it until its degree is depleted: a new vertex of highest degree is selected every time an edge is added.
For example, given the graphical degree sequence $2, 2, 2, 1, 1$, the algorithm proceeds as follows:
which yields the simple path $P_5$ on five vertices (where I broke the ties selecting the leftmost highest or smallest).
Is there a counterexample for which this algorithm does not yield a graph realization?
NOTE: I expect the answer to be affirmative, since this algorithm won't ever connect two vertices of low degree unless it depleted everything else. Therefore, if any graph realization of a certain degree sequence must contain an edge between two vertices of low degree we're done, but I wasn't able to find such an example.