stumpnum

Stumpnum is a graph-theoretic invariant defined as the minimum number of rounds required to prune all leaves from a finite undirected graph by simultaneously removing all vertices of degree 1 in each round. In each round, every leaf (a vertex of degree 1) is deleted along with its incident edge; the degrees of neighboring vertices are updated, and the process is repeated. If the graph initially has minimum degree at least 2, its stumpnum is 0; if the pruning eliminates the entire graph, the stumpnum equals the number of rounds taken to reach emptiness.

Examples illustrate its behavior. A cycle graph C_n has stumpnum 0, since there are no leaves to remove. A path graph P_n has stumpnum equal to ceil(n/2) in general, because each round removes up to two extreme vertices. A star graph S_m = K1,m has stumpnum 1, since removing all leaves leaves a single center vertex (which then has degree 0 and is not a leaf in the next round).

Computation and interpretation. The stumpnum is related to, but distinct from, core decompositions and k-cores. It provides a coarse measure of peripheral structure in a graph and can be viewed as a pruning depth. In practice, it can guide preprocessing steps for network analysis or visual simplification by indicating how many rounds of leaf removal are needed to reach a more central subgraph.

See also: leaf-stripping, core decomposition, k-core, degeneracy.