Bounded degree spanning trees
Given a connected graph G, let a ΔT-spanning tree of G be a spanning tree of G of maximum degree bounded by ΔT. It is well known that for each ΔT ≥ 2 the problem of deciding whether a connected graph has a ΔT-spanning tree is NP-complete. In this paper we investigate this problem when additionally connectivity and maximum degree of the graph are given. A complete characterization of this problem for 2- and 3-connected graphs, for planar graphs, and for ΔT = 2 is provided.
Our first result is that given a biconnected graph of maximum degree 2ΔT - 2, we can find its ΔT-spanning tree in time O(m + n3/2). For graphs of higher connectivity we design a polynomial-time algorithm that finds a ΔT-spanning tree in any k-connected graph of maximum degree k(ΔT − 2) + 2. On the other hand, we prove that deciding whether a k-connected graph of maximum degree k(ΔT - 2) + 3 has a ΔT-spanning tree is NP-complete, provided k ≤ 3. For arbitrary k ≥ 3 we show that verifying whether a k-connected graph of maximum degree k(ΔT - 1) has a ΔT-spanning tree is NP-complete. In particular, we prove that the Hamiltonian path (cycle) problem is NP-complete for k-connected k-regular graphs, if k > 2. This extends the well known result for k = 3 and fully characterizes the case ΔT = 2.
For planar graphs it is NP-complete to decide whether a k-connected planar graph of maximum degree ΔG has a ΔT-spanning tree for k = 1 and Δg > ΔT ≥ 2, for k = 2 and ΔG > 2(ΔT -1) ≥ 2, and for k = 3 and ΔG > ΔT = 2. On the other hand, we show how to find in polynomial (linear or almost linear) time a ΔT-spanning tree for all other parameters of k, ΔG, and ΔT.
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