On Spanning 2-trees in a Graph

A k-tree is either a complete graph on k vertices or a graph T that contains a vertex whose neighbourhood in T induces a complete graph on k vertices and whose removal results in a k-tree. A subgraph of a graph is a spanning k-tree if it is a k-tree and contains every vertex of the graph. This paper is concerned with spanning 2-trees in a graph. It is shown that spanning 2-trees have close connections with two special types of spanning trees: locally-connected spanning trees (A locally-connected spanning tree of a graph G is a spanning tree such that for every vertex v of T the neighbourhood of v in T induces a connected subgraph in G) and tree 2-spanners (A tree 2-spanner of a graph G is a spanning tree such that for every edge of G not in T the distance in T between the two ends of the edge is two). An approximation algorithm is presented for nding a minimum-weight spanning 2-tree in a weighted complete graph, whose asymptotic performance ratio is at most 2 when edge weights satisfy the triangle inequality, and at most (3 + 4 p 3)=6 1:655 when the graph is a complete Euclidean graph on a set of points in the plane. It is also shown that for any two xed integers k > k 0 1, it is NP-complete to determine, given a graph G and a spanning k 0-tree T of G, whether G has a spanning k-tree that contains T .