Some New Results for k-Dense Trees: NP-Completeness Theory

Franceschini, Gianni (2001) Some New Results for k-Dense Trees: NP-Completeness Theory. Technical Report del Dipartimento di Informatica . Università di Pisa, Pisa, IT.

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Abstract

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 consider some NP-complete problems in the area of Graph Theory. For each of these problems we study the restriction to the class of complete k-dense trees, for a generic integer k, and we try to establish whether there are some values of k for which the restriction of the original problem is in P and some other values of k for which this restriction still remains in the class of NP-complete problems.

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