better and probably more appropriate, we would like to keep our term “double-threshold” in this paper.

#### Other generalizations of threshold graphs. {#other-generalizations-of-threshold-graphs. .unnumbered}

There are many other generalizations of threshold graphs such as threshold signed graphs [@BenzakenHW85], threshold tolerance graphs [@MonmaRT88], quasi-threshold graphs (also known as trivially perfect graphs) [@YanCC96], weakly threshold graphs [@Barrus18], and paired threshold graphs [@RavanmehrPBM18]. We omit the definitions of these graph classes and only note that some small graphs show that these classes are incomparable to the class of double-threshold graphs (e.g., $3K_{2}$ and bull for threshold signed graphs, $2K_{2}$ and bull for threshold tolerance graphs, $C_{4}$ and $2 K_{3}$ for quasi-threshold graphs, $2K_{2}$ and bull for weakly threshold graphs, $C_{4}$ and bull for paired threshold graphs[^2]). For the class of bithreshold graphs introduced by Hammer and Mahadev [@HammerM85] (not the one introduced by Jamison and Sprague [@JamisonS19]), we can use $3K_{2}$ and bull to show that this class is incomparable to the class of double-threshold graphs. It is known that $3K_{2}$ is not a bithreshold graph [@HammerMP93], while $3K_{2}$ is a double-threshold graph as we will show later (Lemma \[lem:biperm-&gt;dtg\]). We will also show later that bull is not a double-threshold graph (Lemma 