If the event signals the death of a semi-square, it has to be deleted from every clique C it is an element of. If any of these cliques has been increased by the addition of a semi-square since the last deletion of a semi-square, it is a maximal clique and has to be printed out before the semi-square is deleted. Else, the semi-square is just deleted from C, without printing out the clique. A deletion can yield a reduced clique that is not unique anymore, or that is only a subset of some other clique in Q. Thus, for every clique in which some semi-square has been deleted, one has to verify its uniqueness and extensibility, and it will be deleted from Q if it is not unique or if it is extendable. Note that after checking these two properties, the y-structure invariance holds, because the y-structure just holds all maximal cliques of active semi-squares. Thus, not every maximal clique in the y-structure is a maximal clique with respect to the whole set of semi-squares.