Quasi-threshold graphs are defined recursively by the following rules:
1.
(1) K1 is a quasi-threshold graph,
2.
(2) adding a new vertex adjacent to all vertices of a quasi-threshold graph results in a quasi-threshold graph,
3.
(3) the disjoint union of two quasi-threshold graphs is a quasi-threshold graph.
This paper gives some new equivalent definitions of a quasi-threshold graph. From them, linear time recognition algorithms follow. We also give linear time algorithms for the edge domination problem and the bandwidth problem in this class of graphs.
