fouradjacent

Fouradjacent, also written as four-adjacent or four-connectivity, refers to a relationship between cells in a two-dimensional square lattice in which a cell is considered adjacent to its four orthogonally neighboring cells: up, down, left, and right. This concept is central in grid graphs, digital topology, and image processing, where connectivity and neighborhood definitions govern how regions are explored and labeled.

Formally, for a grid coordinate (x, y), the four-adjacent neighbors are (x−1, y), (x+1, y), (x, y−1), and (x, y+1). In a finite grid, boundary cells have fewer than four neighbors. Four-adjacency contrasts with eight-adjacency (or 8-connectivity), which also includes diagonal neighbors (x±1, y±1). The degree of a typical interior cell under four-adjacency is four, while boundary and corner cells have fewer.

In graph terms, a grid with four-adjacency is a graph where each node corresponds to a grid cell and edges connect four-adjacent pairs. Algorithms operating on such grids—such as breadth-first search, Dijkstra, A* pathfinding, or flood fill—often restrict movements to the four cardinal directions.

Applications of four-adjacency include image segmentation, connected-component labeling under 4-connectivity, and simulations on lattice models. Understanding the distinction between four- and eight-adjacency is important, as it can affect connectivity results, region labeling, and the behavior of diffusion or pathfinding processes.

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