Elsevier

Discrete Mathematics

Volume 306, Issue 5, 28 March 2006, Pages 469-477
Discrete Mathematics

A degree bound on decomposable trees

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Abstract

A n-vertex graph is said to be decomposable if for any partition (λ1,,λp) of the integer n, there exists a sequence (V1,,Vp) of connected vertex-disjoint subgraphs with |Vi|=λi. In this paper, we focus on decomposable trees. We show that a decomposable tree has degree at most 4. Moreover, each degree-4 vertex of a decomposable tree is adjacent to a leaf. This leads to a polynomial time algorithm to decide if a multipode (a tree with only one vertex of degree greater than 2) is decomposable. We also exhibit two families of decomposable trees: arbitrary large trees with one vertex of degree 4, and trees with an arbitrary number of degree-3 vertices.

Keywords

Tree decomposition
Integer partition
Computational complexity

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