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.^[1]

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]

References

1. Díaz, Josep; Petit, Jordi; Serna, Maria (2002), "A survey of graph layout problems" (PDF), ACM Computing Surveys, 34 (3): 313–356, doi:10.1145/568522.568523.
2. Mehlhorn, Kurt; Sanders, Peter (2008). Algorithms and Data Structures: The Basic Toolbox (PDF). Springer.
3. Cuthill, E.; McKee, J. (1969), "Reducing the bandwidth of sparse symmetric matrices", Proceedings of the 1969 24th national conference (ACM '69), ACM, pp. 157–172, doi:10.1145/800195.805928.
4. George, J. Alan (1977), "Solution of linear systems of equations: direct methods for finite element problems", Sparse matrix techniques (Adv. Course, Technical Univ. Denmark, Copenhagen, 1976), Berlin: Springer, pp. 52–101. Lecture Notes in Math., Vol. 572, MR 0440883.
5. Lipton, Richard J.; Tarjan, Robert E. (1979), "A separator theorem for planar graphs", SIAM Journal on Applied Mathematics, 36 (2): 177–189, doi:10.1137/0136016.