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 = 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 = , 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.
The level structure of a graph can be computed by a variant of breadth-first search:
algorithm level-BFS:
Q ←
for ℓ from 0 to ∞:
process // the set Q holds all vertices at level ℓ
mark all vertices in Q as discovered
Q' ←
for u in Q:
for each edge :
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.