will use in the proof of Theorem \[general\].

A graph $G$ is a threshold graph if and only if $V(G)$ can be partitioned into a clique and an independent set and moreover there exists a labeling $i_1,\dots, i_k$ of the vertices in the independent set such that $N(i_1) \supseteq N(i_2) \supseteq \dots \supseteq N(i_k)$. $\hfill$ $\blacksquare$

Note that it is immediate in this definition to see that the lex and colex graphs are threshold. In the colex case only (at most) one vertex in the independent set has any neighbors at all. In the lex graph case the clique consists (with one possible exception) of dominant vertices, so all the vertices in the independent set are joined either to all the vertices in the clique or all but one (and the one missing vertex is the same in each case). The following equivalent characterization of threshold graphs will help us define a way to move a graph towards being threshold.

\[thresholdnbhd\] A graph $G$ is threshold if and only if for all $x,y\in V(G)$ we have $N[x] \subseteq N[y]$ or $N[y]\subseteq N[x]$. $\hfill$ $\blacksquare$

We now define a compression move that makes a graph “more threshold". We will use this 