Recognizing Intersection Patterns

Publisher Summary Many combinatorial problems have the following form: given an n × n matrix A = ( a ij ) decide whether there are sets S 1 , S 2 , . . . , S n such that | S i S j |= a ij for all choices of i and j . If the answer is affirmative, then A is called an “intersection pattern.” Recognizing intersection patterns does not seem easy—for example, deciding whether there is a projective plane of order ten amounts to deciding whether a certain matrix of size 112 × 112 is an intersection pattern. This chapter discusses the recognizing intersection patterns with a ij = 3 for all i is an NP-complete problem. In a sense, the bound a ij ≤ 3 is as severe as one can impose and still expect NP-completeness: recognizing intersection patterns with a ij = 2 for all i amounts to recognizing line-graphs, which is known to be easy. It is convenient to represent each would-be intersection pattern A = (a ij ) with a ii = 3 for all i by a multigraph H in which every two distinct vertices w i , w j are joined by precisely a ij edges. By an admissible partition of H , it is shown in the chapter that partition of its edge-set into disjoint cliques such that every vertex belongs to at most three of these cliques.