This paper proposes an axiomatic for Cyclic Foam Topological Field theories. That is Topological Field theories, corresponding to String theories, where particles are arbitrary graphs. World surfaces in this case are two-manifolds with one-dimensional singularities. We proved that Cyclic Foam Topological Field theories one-to-one correspond to graph-Cardy-Frobenius algebras, that are families $(A,B_\star,��)$, where $A=\{A^s|s\in S\}$ are families of commutative associative Frobenius algebras, $B_\star = \bigoplus_{��\in��} B_��$ is an graduated by graphes, associative algebras of Frobenius type and $��=\{��_��^s: A^s\to (B_��)|s\in S,��\in ��\}$ is a family of special representations. There are constructed examples of Cyclic Foam Topological Field theories and its graph-Cardy-Frobenius algebras

14 pages