Information Processing Letters 89 (2004) 181–185 www.elsevier.com/locate/ipl Partitioning of trees for minimizing height and cardinality András Kovács a,b , Tamás Kis b,∗ a Budapest University of Technology, Magyar tudósok körútja 2/d, 1117 Budapest, Hungary b Computer and Automation Research Institute, Kende utca 13–17, 1111 Budapest, Hungary Received 11 February 2003; received in revised form 3 November 2003 Communicated by S. Albers Keywords: Partitioning of trees; Graph algorithms; Analysis of algorithms; Computational complexity 1. Introduction Partitioning a tree T = (V,E) into q subtrees P = {ST 1 ,..., ST q } such that a given set of constraints is satisfied and a criterion is optimized constitutes a widely studied class of problems that has numerous applications, see, e.g., [1–5]. In this note we consider new criteria and provide polynomial time algorithms. Before presenting our results, we introduce the nota- tion and terminology used throughout the paper. In the sequel T = (V,E) always denotes a rooted tree with vertex-set V , edge-set E, and root r . The sons of a vertex v ∈ V will be denoted by S(v), noting that S(v) =∅ if and only if v is a leaf. Let T (v) be the subtree of T rooted at v consisting of v and all vertices down to the leaves. The following definitions apply to T and also to all T (v). P ={ST 1 ,..., ST q } is a partitioning of T if and only if (a) each component ST i is a subtree (connected subgraph) of T , (b) the ST i are disjoint, and * Corresponding author. E-mail addresses: akovacs@mit.bme.hu (A. Kovács), tamas.kis@sztaki.hu (T. Kis). (c) the union of the vertex-sets V(ST i ) of the ST i equals V . The cardinality of a partitioning P of T is defined as q(P) =|P |. Each ST i is rooted at the vertex closest to r in T . The root component of P is the one containing r , and will be denoted by ST r . The root component weight of P is the total weight of the vertices in ST r , i.e., rw(P ) = ∑ u∈ST r w(u). For any partitioning P of T , let T P denote the rooted tree obtained from T by contracting each ST i ∈ P into a vertex. The height h(T ) of a rooted tree T is the maximum number of edges of paths having one end at the root. The height h(P ) of a partitioning P is the height of T P . The level ℓ P (v) of a vertex v ∈ V with respect to a partitioning P of T is the height of the induced partitioning P ′ of T (v) consisting of those ST i ∈ P with V(ST i ) ⊆ V (T (v)) and also ST v , where ST v = ST j ∩ T (v) and v ∈ ST j ∈ P . Given a non-negative weight function w : V → R + on the vertices of T and a constant W , we say that a partitioning P ={ST 1 ,..., ST q } of T satisfies the knapsack constraint if and only if  v∈V(ST i ) w(v)  W for every ST i ∈ P. (1) 0020-0190/$ – see front matter  2003 Elsevier B.V. All rights reserved. doi:10.1016/j.ipl.2003.11.004