Determining X-chains in graph states

The representation of graph states in the X-basis as well as the calculation of graph state overlaps can efficiently be performed by using the concept of X-Chains [Phys. Rev. A 92(1) 012322]. We present a necessary and sufficient criterion for X-chains and show that they can efficiently be determined by Bareiss algorithm. An analytical approach for searching X-chain groups of a graph state is proposed. Furthermore we generalize the concept of X-chains to so-called Euler chains, whose induced subgraphs are Eulerian. This approach helps to determine if a given vertex set is an X-chain and we show how Euler chains can be used in the construction of multipartite Bell inequalities for graph states.