On exact blockers and anti-blockers, Δ-conjecture, and related problems

Let us consider two binary systems of inequalities (i) Cx ≥ e and (ii) Cx ≤ e , where C ∈ {0, 1}m×n is an m × n (0, 1)-matrix, x ∈ {0, 1}n, and e is the vector of m ones. The set of all support-minimal (respectively, supportmaximal) solutions x to (i) (respectively, to (ii)) is called the blocker (respectively, anti-blocker). A blocker B (respectively, anti-blocker A) is called exact if Cx = e for every x ∈ B (respectively, x ∈ A). Exact blockers can be completely characterized. There is a one-to-one correspondence between them and P4-free graphs (along with a well known one-to-one correspondence between the latter and the so-called readonce Boolean functions). However, the class of exact anti-blockers is wider and more sophisticated. We demonstrate that it is closely related to the so-called CIS graphs, more general `-CIS d-graphs, and ∆-conjecture.