A special configuration of graph state stabilizers, which contains only Pauli σX operators, is studied. The vertex sets ξ associated with such configurations are defined as what we call X chains of graph states. The X chains of a general graph state can be determined efficiently. They form a group structure such that one can obtain the explicit representation of graph states in the X basis via the so-called X-chain factorization diagram. We show that graph states with different X-chain groups can have different probability distributions of X-measurement outcomes, which allows one to distinguish certain graph states with X measurements. We provide an approach to find the Schmidt decomposition of graph states in the X basis. The existence of X chains in a subsystem facilitates error correction in the entanglement localization of graph states. In all of these applications, the difficulty of the task decreases with increasing number of X chains. Furthermore, we show that the overlap of two graph states can be efficiently determined via X chains, while its computational complexity with other known methods increases exponentially.
