In this talk I present work in progress by my FNRS MIS funded research team (Pilar Terres, Pierre Saint-Germier, Joao Daniel Dantas) working on explanatory inference. We came up with a criterion for relevance of the entailment relation, relative to a given logic. One of the weak criteria of relevance presented in the literature is the principle of variable sharing: if a (multiple conclusion) sequent is relevantly valid then every formula in the sequent needs to have at least one variable in common with the other formulas in the sequent. I present a couple of cases from which it should be clear that this criterion (while being necessary) certainly is not sufficient for relevance. We solve these problems by analyzing relevance in terms of connectivity. The idea is to say that a sequent is relevantly valid iff a connected graph (of a specific nature) can be established that contains all of the formulas of the sequent. The basis of this idea is the concept of a constitution of a logic. This is a set of sequents that express full logical grounds of all formulas of the language of the logic (the grounded formula of each sequent is underlined, the non-underlined formulas are the partial grounds--examples are “A,B>A&B”, “A&B>A” and “A, A->B > B” in the case of classical logic). The partial grounds (the non-underlined formulas) of each formula determine the way in which formulas of the potentially relevant sequents can be connected to other formulas of the sequent. We will present and motivate the criterion, give a couple of examples, and present some graph-theoretical results concerning this criterion.