On universal positive graphs

We study the existence of the universal computable numberings and the universal graphs for various classes of positive graphs. It is known that each $\forall$-axiomatizable class of graphs $K$ can be characterized as follows: A graph $G$ belongs to $K$ if and only if for a given family of finite graphs $\mathbf{F}$ no graph in $\mathbf{F}$ is isomorphically embeddable into $G$. If all graphs in $\mathbf{F}$ are weakly connected; then, under additional effectiveness conditions, the corresponding class $K$ has some universal computable numbering and universal positive graph. The effectiveness conditions hold for forests, bipartite graphs, planar graphs, and $n$-colorable graphs (for a fixed number $n$). If $\mathbf{F}$ is a finite family of the graphs with weakly connected complement then the corresponding class $K$ contains a universal positive graph (in general, a universal computable numbering for $K$ may fail to exist).