pairsessential

Pairsessential is a concept in graph theory and data analysis that refers to a minimal subset of pairwise relations on a finite set chosen to preserve a specified global property of the original dataset. It is used to simplify complex networks while maintaining essential structure.

In its common form, pairsessential concerns preserving connectivity in an undirected graph. Given a graph G = (V, E), a subset E' ⊆ E is pair-essential if the graph G' = (V, E') has the same set of connected components as G. Such a subset is minimal with respect to inclusion.

Equivalently, a pair-essential edge set is a spanning forest of G: it connects all vertices within each component but uses as few edges as possible. For a graph with c connected components and n vertices, any pair-essential set contains exactly n − c edges, and removing any edge would split a component.

Computing a pair-essential set can be done with standard graph-sparsification techniques. A spanning forest is produced by algorithms such as Kruskal’s or Prim’s, or by simple depth-first searches component by component. The process runs in near-linear time relative to the size of the graph.

Applications include network design, data compression, and visualization, where preserving the essential connectivity is desirable while reducing edge count. The concept also extends to directed graphs, where one might seek a pair-essential subset that preserves reachability or strongly connected components, depending on the target property.

See also: spanning tree, spanning forest, graph sparsification.