One of the common obstacles for learning causal models from data is that high-order conditional independence (CI) relationships between random variables are difficult to estimate. Since CI tests with conditioning sets of low order can be performed accurately even for a small number of observations, a reasonable approach to determine casual structures is to base merely on the low-order CIs. Recent research has confirmed that, e.g. in the case of sparse true causal models, structures learned even from zero- and first-order conditional independencies yield good approximations of the models. However, a challenging task here is to provide methods that faithfully explain a given set of low-order CIs. In this paper, we propose an algorithm which, for a given set of conditional independencies of order less or equal to $k$, where $k$ is a small fixed number, computes a faithful graphical representation of the given set. Our results complete and generalize the previous work on learning from pairwise marginal independencies. Moreover, they enable to improve upon the 0-1 graph model which, e.g. is heavily used in the estimation of genome networks.