# 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.

## Key words

Graph algorithm panning*K*-tree approximation algorithm NP-completeness

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