Graphs and complete intersection toric ideals

Our purpose is to study the family of simple undirected graphs whose toric ideal is a complete intersection from both an algorithmic and a combinatorial point of view. We obtain a polynomial time algorithm that, given a graph $G$, checks whether its toric ideal $P_G$ is a complete intersection or not. Whenever $P_G$ is a complete intersection, the algorithm also returns a minimal set of generators of $P_G$. Moreover, we prove that if $G$ is a connected graph and $P_G$ is a complete intersection, then there exist two induced subgraphs $R$ and $C$ of $G$ such that the vertex set $V(G)$ of $G$ is the disjoint union of $V(R)$ and $V(C)$, where $R$ is a bipartite ring graph and $C$ is either the empty graph, an odd primitive cycle, or consists of two odd primitive cycles properly connected. Finally, if $R$ is $2$-connected and $C$ is connected, we list the families of graphs whose toric ideals are complete intersection.