Abstract
We consider the degree-preserving spanning tree (DPST) problem: given a connected graph G, find a spanning tree T of G such that as many vertices of T as possible have the same degree in T as in G. This problem is a graph-theoretical translation of a problem arising in the system-theoretical context of identifiability in networks, a concept which has applications in e.g., water distribution networks and electrical networks. We show that the DPST problem is NP-complete, even when restricted to split graphs or bipartite planar graphs. We present linear time approximation algorithms for planar graphs of worst case performance ratio 1−ε for every constant ε > 0. Furthermore we give exact algorithms for interval graphs (linear time), graphs of bounded treewidth (linear time), cocomparability graphs (O(n 4)), and graphs of bounded asteroidal number.
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References
Aaron, M. and M. Lewinter, 0-deficient vertices of spanning trees, NY Acad. Sci. Graph Theory Notes XXVII, (1994), pp. 31–32.
Arnborg S., J. Lagergren and D. Seese, Easy problems for tree-decomposable graphs, Journal of Algorithms 12, (1991), pp. 308–340.
Baker, B. S., Approximation algorithms for NP-complete problems on planar graphs, J. ACM 41, (1994), pp. 153–180.
Bellare, M., O. Goldreich and M. Sudan, Free bits, PCPs and non-approximability — towards tight results, SIAM J. Comput. 27 (1998), pp. 804–915.
Camerini, P. M., G. Galbiati and F. Maffioli, Complexity of spanning tree problems: Part I, Eur. J. Oper. Res. 5, (1980), pp. 346–352.
Camerini, P. M., G. Galbiati and F. Maffioli, The complexity of weighted multiconstrained spanning tree problems, Colloq. Math. Soc. Janos Bolyai 44, (1984), pp. 53–101.
Damaschke, P., Degree-preserving spanning trees and coloring bounded degree graphs, Manuscript 1997.
Dell'Amico, M., M. Labbé and F. Maffioli, Complexity of spanning tree problems with leaf-dependent objectives, Networks 27, (1996), pp. 175–181.
Garey, M. R. and D.S. Johnson, Computers and Intractability: A guide to the Theory of NP-completeness, Freeman, New York, 1979.
Golumbic, M. C., Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.
Håstad, J., Clique is hard to approximate within n 1-ɛ, Proc. 37th Ann. IEEE Symp. on Foundations of Comput. Sci., (1996), IEEE Computer Society, pp. 627–636.
Lewinter, M., Interpolation theorem for the number of degree-preserving vertices of spanning trees, IEEE Trans. Circ. Syst. CAS-34, (1987), 205.
Lewinter, M. and M. Migdail-Smith, Degree-preserving vertices of spanning trees of the hypercube, NY Acad. Sci. Graph Theory Notes XIII, (1987), 26–27.
Pothof, I. W. M. and J. Schut, Graph-theoretic approach to identifiability in a water distribution network, Memorandum 1283, Faculty of Applied Mathematics, University of Twente, Enschede, the Netherlands, (1995).
Rahal, A co-tree flows formulation for steady state in water distribution networks, Adv. Eng. Softw. 22, (1995), pp. 169–178.
Walter, E., Identifiability of state space models with applications to transformation systems, Springer-Verlag, New York NY, USA, 1982.
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Broersma, H. et al. (1998). Degree-preserving forests. In: Brim, L., Gruska, J., Zlatuška, J. (eds) Mathematical Foundations of Computer Science 1998. MFCS 1998. Lecture Notes in Computer Science, vol 1450. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0055822
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DOI: https://doi.org/10.1007/BFb0055822
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