Dense trees: a new look at degenerate graphs

https://doi.org/10.1016/j.jda.2005.12.008Get rights and content
Under an Elsevier user license
open archive

Abstract

Dense trees are undirected graphs defined as natural extensions of trees. They are already known in the realm of graph coloring under the name of k-degenerate graphs. For a given integer k1, a k-dense cycle is a connected graph, where the degree of each vertex is greater than k. A k-dense forest F=(V,E) is a graph without k-dense cycles as subgraphs. If F is connected, then is a k-dense tree. 1-dense trees are standard trees. We have |E|k|V|k(k+1)/2. If equality holds F is connected and is called a maximal k-dense tree. k-trees (a subfamily of triangulated graphs) are special cases of maximal k-dense trees.

We review the basic theory of dense trees in the family of graphs and show their relation with k-trees. Vertex and edge connectivity is thoroughly investigated, and the role of maximal k-dense trees as “reinforced” spanning trees of arbitrary graphs is presented. Then it is shown how a k-dense forest or tree can be decomposed into a set of standard spanning trees connected through a common “root” of k vertices. All sections include efficient construction algorithms. Applications of k-dense trees in the fields of distributed systems and data structures are finally indicated.

Keywords

k-dense tree
k-dense forest
k-elimination
k-degenerate graph
k-tree
Spanning tree
Connectivity
Decomposition
Interconnection structures

Cited by (0)