Level structure



In the mathematical subfield of graph theory a level structure of an undirected graph is a partition of the vertices into subsets that have the same distance from a given root vertex.

Definition and construction
Given a connected graph G = (V, E) with V the set of vertices and E the set of edges, and with a root vertex r, the level structure is a partition of the vertices into subsets Li called levels, consisting of the vertices at distance i from r. Equivalently, this set may be defined by setting L0 = {r}, and then, for i > 0, defining Li to be the set of vertices that are neighbors to vertices in Li &minus; 1 but are not themselves in any earlier level.

The level structure of a graph can be computed by a variant of breadth-first search:

algorithm level-BFS(G, r): Q ← {r} for ℓ from 0 to ∞: process(Q, ℓ) // the set Q holds all vertices at level ℓ mark all vertices in Q as discovered Q' ← {} for u in Q:            for each edge (u, v): if v is not yet marked: add v to Q'        if Q' is empty: return Q ← Q'

Properties
In a level structure, each edge of G either has both of its endpoints within the same level, or its two endpoints are in consecutive levels.

Applications
The partition of a graph into its level structure may be used as a heuristic for graph layout problems such as graph bandwidth. The Cuthill–McKee algorithm is a refinement of this idea, based on an additional sorting step within each level.

Level structures are also used in algorithms for sparse matrices, and for constructing separators of planar graphs.