There is a well-known correspondence between formulas and cographs, which are unoriented graphs containing no chordless paths of length 3. The aim of this talk is to shape a proof system operating on general graphs instead of formulas. In such a proof system, we lose the tree structure of formulas corresponding to cographs. As a consequence, we can no longer use the standard proof theoretical methods relying on the tree structure of formulas: for instance, we are not able to identify the main connective of a formula.

Thanks to graphs modular decomposition and some techniques from deep inference, we are able to define a proof system with admissible cut rule and which is a conservative extension of MLL with mix.

This talk is based on a joint work with Ross Horne and Lutz Strassburger.