Bi-Squashing S2,2-Designs into (K4 − e)-Designs

A double-star Sq1,q2 is the graph consisting of the union of two stars, K1,q1 and K1,q2, together with an edge joining their centers. The spectrum for Sq1,q2-designs, i.e., the set of all the n∈N such that an Sq1,q2-design of the order n exists, is well-known when q1=q2=2. In this article, S2,2-designs satisfying additional properties are investigated. We determine the spectrum for S2,2-designs that can be transformed into (K4−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 S2,2-designs that can be purely bi-squashed into cyclic (K4−e)-designs (the middle bull designs are also cyclic).