Abstract
Given a connected graph G, let a Δ T-spanning tree of G be a spanning tree of G of maximum degree bounded by Δ T. It is well known that for each Δ T ≥ 2 the problem of deciding whether a connected graph has a Δ T-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 Δ T = 2 is provided.
Our first result is that given a biconnected graph of maximum degree 2Δ T - 2, we can find its Δ T-spanning tree in time O(m + n 3/2). For graphs of higher connectivity we design a polynomial-time algorithm that finds a Δ T-spanning tree in any k-connected graph of maximum degree k(Δ T − 2) + 2. On the other hand, we prove that deciding whether a k-connected graph of maximum degree k(Δ T - 2) + 3 has a Δ T-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(Δ T - 1) has a Δ T-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 Δ T = 2.
For planar graphs it is NP-complete to decide whether a k-connected planar graph of maximum degree Δ G has a Δ T-spanning tree for k = 1 and Δ g > Δ T ≥ 2, for k = 2 and Δ G > 2(Δ T -1) ≥ 2, and for k = 3 and Δ G > Δ T = 2. On the other hand, we show how to find in polynomial (linear or almost linear) time a Δ T-spanning tree for all other parameters of k, Δ G, and Δ T.
Partially supported by EU ESPRIT Long Term Research Project 20244 (ALCOM-IT), DFG Leibniz Grant Me872/6-1, and DFG Project Me872/7-1.
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© 1997 Springer-Verlag Berlin Heidelberg
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Czumaj, A., Strothmann, WB. (1997). Bounded degree spanning trees. In: Burkard, R., Woeginger, G. (eds) Algorithms — ESA '97. ESA 1997. Lecture Notes in Computer Science, vol 1284. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63397-9_9
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DOI: https://doi.org/10.1007/3-540-63397-9_9
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