It is recognized that block-wise dependence is a key requirement for fragile watermarking schemes to thwart vector quantization attack. It has also been proved that dependence with deterministic or limited context is susceptible to transplantation attack or even simple cover-up attacks. In this work, we point out that traditional nondeterministic block-wise dependence is still vulnerable to cropping attacks and propose a 1-D neighborhood forming strategy to tackle the problem. The proposed strategy is then implemented in our new fragile watermarking scheme, which does not resort to cryptography and requires no a priori knowledge about the image for verification. To watermark the underlying image, the gray scale of each pixel is adjusted by an imperceptible quantity according to the consistency between a key-dependent binary watermark bit and the parity of a bit stream converted from the gray scales of a secrete neighborhood formed with the 1-D strategy. The watermark extraction process works exactly the same as the embedding process, and produces a difference map as output, indicating the authenticity and integrity of the image.