Abstract
Given a complete graph with vertex setX and subsetsX 1,X 2,…,X n , the problem of finding a subgraphG with minimum number of edges such that for everyi=1,2,…,n, G contains a spanning tree onX i , arises in the design of vacuum systems. In general, this problem is NP-complete and it is proved that forn=2 and 3 this problem is polynomial-time solvable. In this paper, we prove that forn=4, the problem is also polynomial-time solvable and give a method to construct the corresponding graph.
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References
Aho, A.V., Hopcroft, J.E. and Ullman, J.D., Data Structures and Algorithms, Addison Wesley Publishing Company, 1983.
Chao, S.C. and Du, D.Z., A sufficient optimality condition for valve-placement problem,J. North-East Heavy Industry Institut,4 (1984). (Chinese)
Du, D.Z., On complexity of subset interconnection designs,Technical Report 91-23 DIMACS, (1991).
___, An optimization problem,Discrete Appl. Math.,14 (1986), 101–104.
Du, D.Z. and Chen, Y.M., Placement of valves in vacuum systems,J. Electric Light Sources,4 (1976). (Chinese)
Du, D.Z. and Miller, Z., Matroids and subset interconnection design,SIAM J. Disc Math.,1 (1988), 416–424.
Tang, T.Z., An optimality condition for minimum feasible graphs,Applied Mathematics,2:2 (1989), 21–24. (Chinese)
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Xu, Y., Fu, X. On the minimum feasible graph for four sets. Appl. Math. 10, 457–462 (1995). https://doi.org/10.1007/BF02662501
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DOI: https://doi.org/10.1007/BF02662501