Graphical virtual links and a polynomial for signed cyclic graphs

For a signed cyclic graph G, we can construct a unique virtual link L by taking the medial construction and converting 4-valent vertices of the medial graph to crossings according to the signs. If a virtual link can occur in this way then we say that the virtual link is graphical. In this paper, we shall prove that a virtual link L is graphical if and only if it is checkerboard colorable. On the other hand, we introduce a polynomial F[G] for signed cyclic graphs, which is defined via a deletion-marking recursion. We shall establish the relationship between F[G] of a signed cyclic graph G and the bracket polynomial of one of the virtual link diagrams associated with G. Finally, we give a spanning subgraph expansion for F[G].