fourconnected

Fourconnected, or 4-connectivity, is a neighborhood relation on a two-dimensional square lattice used in digital geometry and image processing. In a grid, two cells are 4-neighbors if their coordinates differ by exactly one unit in one axis and are equal in the other; the 4-neighborhood of a cell consists of its north, south, east, and west adjacent cells. A set of cells is 4-connected if every pair of cells can be joined by a path through 4-neighbors; a 4-connected component is a maximal 4-connected subset.

Compared with 8-connectivity, which also includes diagonal adjacencies, 4-connectivity can yield different component boundaries. Two pixels touching at a corner can be connected under 8-connectedness but remain separate under 4-connectivity. This distinction matters in image segmentation and connected-component labeling, where algorithms must be defined with a chosen connectivity to avoid ambiguity.

Common implementations use a cross-shaped neighbor stencil. Algorithms for extracting 4-connected components include two-pass connected-component labeling, BFS/DFS traversals, or union-find techniques, typically running in time linear to the number of grid cells. These methods are used in binary image analysis, pathfinding on grids, and cellular automata contexts where Von Neumann (4-neighborhood) versus Moore (8-neighborhood) rules determine neighbor interactions.

In higher dimensions, the 4-connected notion generalizes to 6-connected in 3D spaces (face-adjacent cells). The term fourconnected is sometimes used in programming libraries to denote this 4-neighborhood convention.