Some New Results for k-Dense Trees: Graph Theory

Let k be a positive integer. A k-cycle is a connected graph in which each vertex has degree greater than k. A k-dense forest is a graph for which no subgraph is a k-cycle; if a k-dense forest is connected, then it is k-dense tree. A k-leaf is a vertex of a k-dense forest with degree less than or equal to k. Any k-dense forest has at least one k-leaf. If a k-leaf is removed, the resulting graph is still a k-dense forest. This fact is on the basis of another characterization of k-dense forests which make use of the concept of k-elimination, a particular ordering of removal for the vertices of a k-dense forest. In this paper, we study some new properties of the complete k-dense trees, a subclass of the one of k-dense trees. Such properties reveal some interesting relations between the class of complete k-dense trees and the widely studied class of k-trees