Deformation cones of graphical zonotopes for $K_4$-free graph

In this paper, we compute a triangulation of certain faces of the submodular cone. More precisely, graphical zonotopes are generalized permutahedra, and hence their deformation cones are faces of the submodular cone. We give a triangulation of these faces for graphs without induced complete sub-graph on 4 vertices. We deduce the rays of these faces: Minkowski indecomposable deformations of these graphical zonotopes are segments and triangles. Besides, computer experiments lead to examples of graphs without induced complete sub-graph on 5 vertices, whose graphical zonotopes have high dimensional Minkowski indecomposable deformations.