Criticality of switching classes of reversible 2-structures labeled by an Abelian group

International audienceLet V be a finite vertex set and let (A, +) be a finite abelian group. An A-labeled and reversible 2-structure defined on V is a function g : (V x V) \ {(v, v) : v is an element of V} -> A such that for distinct u, v is an element of V, g(u, v) = -g (v, u). The set of A-labeled and reversible 2-structures defined on V is denoted by L(V, A). Given g is an element of Y(V, A), a subset X of V is a clan of g if for any x, y is an element of X and v is an element of V\X, g(x, v) = g(y, v). For example, empty set, V and {v} (for v E V) are clans of g, called trivial. An element g of Y(V, A) is primitive if V >= 3 and all the clans of g are trivial. The set of the functions from V to A is denoted by,/(V, A). Given g is an element of L(V, A), with each s is an element of J(V, A) is associated the switch g(s) of g by s defined as follows: given distinct x, y E V, gs (x, y) = s(x) g (x, y) s(y). The switching class of g is {g(s) : s is an element of J(V, A)}. Given a switching class G subset of Y(V, A) and X subset of V, {g((XxX)\,(x,x):x is an element of X}) g is an element of 1 G} is a switching class, denoted by G[X]. Given a switching class G subset of L(V, A), a subset X of V is a clan of G if X is a clan of some g is an element of G. For instance, every X subset of V such that min(X, V \ X) <= 1 is a clan of G, called trivial. A switching class G subset of L(V, A) is primitive if V <= 4 and all the clans of G are trivial. Given a primitive switching class G subset of L(V, A), G is critical if for each v is an element of V, G - v is not primitive. First, we translate the main results on the primitivity of A-labeled and reversible 2-structures in terms of switching classes. For instance, we prove the following. For a primitive switching class G subset of L(V, A) such that V >= 8, there exist u, v is an element of V such that u not equal v and G[V \ {u, v}] is primitive. Second, we characterize the critical switching classes by using some of the critical digraphs described in [Y. Boudabous and P. Ille, Indecomposability graph and critical vertices of an indecomposable graph, Discrete Math. 309 (2009) 2839-2846]