Level structure

In the mathematical subfield of graph theory a level structure of a rooted graph is a partition of the vertices into subsets that have the same distance from a given root vertex.^[1]

An example for an undirected Graph with a vertex r and its corresponding level structure

For the concept in algebraic geometry, see level structure (algebraic geometry)

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 L_i 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 L_i to be the set of vertices that are neighbors to vertices in L_i − 1 but are not themselves in any earlier level.^[1]

The level structure of a graph can be computed by a variant of breadth-first search:^[2]: 176

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.^[1]

Applications

The partition of a graph into its level structure may be used as a heuristic for graph layout problems such as graph bandwidth.^[1] The Cuthill–McKee algorithm is a refinement of this idea, based on an additional sorting step within each level.^[3]

Level structures are also used in algorithms for sparse matrices,^[4] and for constructing separators of planar graphs.^[5]