{"abstract":[{"lang":"eng","text":"Given a connected graph \$G\$, let a \$dT\$-spanning tree of \$G\$ be a spanning tree of \$G\$ of maximum degree bounded by \$dT\$. It is well known that for each \$dT ge 2\$ the problem of deciding whether a connected graph has a \$dT\$-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 \$dT=2\$ is provided. Our first result is that given a biconnected graph of maximum degree \$2dT-2\$, we can find its \$dT\$-spanning tree in time \$O(m+n^3/2)\$. For graphs of higher connectivity we design a polynomial-time algorithm that finds a \$dT\$-spanning tree in any \$k\$-connected graph of maximum degree \$k(dT-2)+2\$. On the other hand, we prove that deciding whether a \$k\$-connected graph of maximum degree \$k(dT-2)+3\$ has a \$dT\$-spanning tree is NP-complete, provided \$k le 3\$. For arbitrary \$k ge 3\$ we show that verifying whether a \$k\$-connected graph of maximum degree \$k(dT-1)\$ has a \$dT\$-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 \$dT=2\$. For planar graphs it is NP-complete to decide whether a \$k\$-connected planar graph of maximum degree \$dG\$ has a \$dT\$-spanning tree for \$k=1\$ and \$dG > dT ge 2\$, for \$k=2\$ and \$dG > 2(dT-1) ge 2\$, and for \$k=3\$ and \$dG > dT = 2\$. On the other hand, we show how to find in polynomial (linear or almost linear) time a \$dT\$-spanning tree for all other parameters of \$k\$, \$dG\$, and \$dT\$."}],"user_id":"15415","citation":{"ama":"Czumaj A, Strothmann W-B. Bounded degree spanning trees. In: Proceedings of the Fifth Annual European Symposium on Algorithms (ESA’97). ; 1997. doi:10.1007/3-540-63397-9_9","ieee":"A. Czumaj and W.-B. Strothmann, “Bounded degree spanning trees,” 1997, doi: 10.1007/3-540-63397-9_9.","short":"A. Czumaj, W.-B. Strothmann, in: Proceedings of the Fifth Annual European Symposium on Algorithms (ESA’97), 1997.","chicago":"Czumaj, Artur, and Willy-Bernhard Strothmann. “Bounded Degree Spanning Trees.” In Proceedings of the Fifth Annual European Symposium on Algorithms (ESA’97), 1997. https://doi.org/10.1007/3-540-63397-9_9.","apa":"Czumaj, A., & Strothmann, W.-B. (1997). Bounded degree spanning trees. Proceedings of the Fifth Annual European Symposium on Algorithms (ESA’97). https://doi.org/10.1007/3-540-63397-9_9","bibtex":"@inproceedings{Czumaj_Strothmann_1997, title={Bounded degree spanning trees}, DOI={10.1007/3-540-63397-9_9}, booktitle={Proceedings of the Fifth Annual European Symposium on Algorithms (ESA’97)}, author={Czumaj, Artur and Strothmann, Willy-Bernhard}, year={1997} }","mla":"Czumaj, Artur, and Willy-Bernhard Strothmann. “Bounded Degree Spanning Trees.” Proceedings of the Fifth Annual European Symposium on Algorithms (ESA’97), 1997, doi:10.1007/3-540-63397-9_9."},"publication":"Proceedings of the Fifth Annual European Symposium on Algorithms (ESA'97)","language":[{"iso":"eng"}],"author":[{"full_name":"Czumaj, Artur","first_name":"Artur","last_name":"Czumaj"},{"full_name":"Strothmann, Willy-Bernhard","first_name":"Willy-Bernhard","last_name":"Strothmann"}],"date_updated":"2022-01-06T06:54:14Z","department":[{"_id":"63"}],"date_created":"2020-10-05T07:13:42Z","_id":"19869","type":"conference","year":"1997","title":"Bounded degree spanning trees","publication_status":"published","doi":"10.1007/3-540-63397-9_9","publication_identifier":{"issn":["0302-9743","1611-3349"],"isbn":["9783540633976","9783540695363"]},"status":"public"}