Spanning 2-trees

Abstract

A k-tree is defined recursively to be either a K-clique or a graph T that contains a vertex v whose neighbourhood in T induces a k-clique and whose removal results in a k-tree. The existence of a spanning k-tree in a communication network is closely related to the reliability of the network, and it is known that the problem of determining whether a graph contains a spanning k-tree is NP-complete for any fixed k ≥2. In this paper, several sufficient conditions are given for the existence of spanning 2-trees in a graph. An approximation algorithm is presented for finding a spanning 2-tree with minimum weight in a weighted complete graph. The asymptotic performance ratio of the algorithm is 2 when edge weights satisfy the triangle inequality, and 1.655 when the graph is a complete Euclidean graph on a set of points in the plane. It is also shown that it is NP-complete to determine whether a graph admits a spanning 2-tree that contains a given spanning tree.

This work was partially supported by a Direct Grant for Research from the Chinese University of Hong Kong and an Earmarked Research Grant from the Research Grants Council of Hong Kong.