Unitsnodesconnected

Unitsnodesconnected is a term used in geometry and network theory to describe the connectivity properties of a unit-distance graph derived from a finite set of points. In this context, one constructs a graph by placing a vertex at each point and adding an edge between two vertices whenever the corresponding points are exactly one unit apart.

Formally, let P be a finite set of points in the plane or in Euclidean space R^d. The unit-distance graph G = (P, E) has edges E = { {p, q} : dist(p, q) = 1 }. The configuration P is said to be unitsnodesconnected if the graph G is connected, meaning there is a path between every pair of vertices. In some formulations, the term may also refer to the size of the largest connected component of G, denoted by the maximum component size or UNC(P), or to a boolean indicator of connectedness.

Examples illustrate the idea: a square with side length 1 yields a unit-distance graph that is connected; similarly, an equilateral triangle of side length 1 yields a complete, connected graph. In contrast, a set of points where isolated clusters are separated by more than unit distance produces a unit-distance graph with multiple components, hence not unitsnodesconnected.

Computationally, determining unitsnodesconnected involves computing pairwise distances to identify unit edges and then performing a connectivity test such as breadth-first search or union-find. The typical complexity is O(n^2) to build edges plus near-linear time to test connectivity, where n is the number of points. The concept is relevant in discrete and computational geometry, sensor networks, and the study of unit-distance graphs. Related ideas include unit disk graphs (distance ≤ 1) and problems in graph realization and rigidity.