baumdominant

Baumdominant is a term used in graph theory and network analysis to describe a central vertex in a tree-like structure. Derived from the German word Baumen meaning tree, the term conveys the idea of dominance in the sense of balancing the tree around a single point. A baumdominant node is the tree’s centroid in weight terms and is central to algorithms that require balanced decomposition.

Definition: Let T be a finite tree with total weight W, where each vertex v has nonnegative weight w(v) and W = sum w(v). A vertex v is baumdominant if, after removing v, the resulting components each have total weight at most W/2. Equivalently, the maximum component weight after removal of v is at most W/2. The baumdominant set consists of all vertices satisfying this property; in a tree this set is either a single vertex or two adjacent vertices.

Computation: A linear-time algorithm performs a postorder traversal to compute subtree weights s(v) = sum of weights in the subtree rooted at v. For each vertex v, compute max_component = max( max over children c of s(c), W - s(v) ). If max_component <= W/2, v is baumdominant.

Properties and relations: Baumdominant nodes coincide with tree centroids when weights are vertex weights. The concept generalizes to edge-weighted trees by substituting edge weights into the component sums. The baumdominant set minimizes the maximum distance to other parts of the tree in some variants and often serves as a natural breakpoint for balanced partitioning.

Applications: Used in load balancing, network design, parallel processing, and data clustering where balanced decomposition of a tree is desired.

See also: centroid, tree center, tree decomposition.