# 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 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 − 1 but are not themselves in any earlier level.[1]

If a breadth first search of G is performed, starting from r, then the vertices in each level will be found as a consecutive subsequence of the breadth first ordering of the graph. These subsets may be computed by performing a breadth first search, calculating the level of each vertex v as it is processed by the search by adding one to the minimum level of an already-processed neighbor of v, and storing this level with v so that its later neighbors may perform the same calculation.

## 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.[2]

Level structures are also used in algorithms for sparse matrices,[3] and for constructing separators of planar graphs.[4]

## References

1. ^ a b c d Díaz, Josep; Petit, Jordi; Serna, Maria (2002), "A survey of graph layout problems", ACM Computing Surveys 34 (3): 313–356, doi:10.1145/568522.568523.
2. ^ 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.
3. ^ 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.
4. ^ 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.