Bi-Squashing S 2,2 -Designs into ( K 4 − e )-Designs

A double-star S q 1 , q 2 is the graph consisting of the union of two stars, K 1 , q 1 and K 1 , q 2 , together with an edge joining their centers. The spectrum for S q 1 , q 2 -designs, i.e., the set of all the n ∈ N such that an S q 1 , q 2 -design of the order n exists, is well-known when q 1 = q 2 = 2 . In this article, S 2 , 2 -designs satisfying additional properties are investigated. We determine the spectrum for S 2 , 2 -designs that can be transformed into ( K 4 − e ) -designs by a double squash (bi-squash) passing through middle designs whose blocks are copies of a bull (the graph consisting of a triangle and two pendant edges). Here, the use of the difference method enables obtaining cyclic decompositions and determining the spectrum for cyclic S 2 , 2 -designs that can be purely bi-squashed into cyclic ( K 4 − e ) -designs (the middle bull designs are also cyclic).