The critical ideals of a graph are the determinantal ideals of the
generalized Laplacian matrix associated to a graph. In this article we provide
a set of minimal forbidden graphs for the set of graphs with at most three
trivial critical ideals. Then we use these forbidden graphs to characterize the
graphs with at most three trivial critical ideals and clique number equal to 2
and 3.Comment: 33 pages, 3 figure