On the SPANNING k-TREE problem

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Abstract

A k-tree T is defined recursively as being either a clique of size k or a graph having a vertex x whose neighbourhood is a clique of size k and such that T−x is a k-tree. We consider the following “SPANNING k-TREE” problem: Given a graph G, does G possess a subgraph which is a k-tree and which contains every vertex of G? This problem is known to be NP-complete. We prove that it remains NP-complete when the input is restricted to the classes of split graphs or graphs with maximum degree 3k+2. We show that the SPANNING 2-TREE problem remains NP-complete when restricted to planar graphs with maximum degree at most 6. We present polynomial-time algorithms that solve the spanning k-tree problem when the input graph is a split-comparability graph or an interval graph.

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