ON THE MINIMUM FEASIBLE GRAPH FOR FOUR SETS

ＯＮＴＨＥＭＩＮＩＭＵＭＦＥＡＳＩＢＬＥＧＲＡＰＨＦＯＲＦＯＵＲＳＥＴＳＸＵＹＩＮＦＥＮＧＡＮＤＦＵＸＩＡＯＢＩＮＧＡｂｓｔｒａｃｔ：ＧｉｖｅｎａｃｏｍｐｌｅｔｅｇｒａｐｈｗｉｔｈｖｅｒｔｅｘｓｅｔＸａｎｄｓｕｂｓｅｔｓＸ＿１，Ｘ＿２，．．．，Ｘ＿ｎ...

On the minimum feasible graph for four sets

Given a complete graph with vertex set X and subsets X_1, X_2,..., X_n, the problem of finding a subgraph G with minimum number of edges such that for every i = 1, 2,..., n G contains a spanning tree on X_i, arises in the design of vaccum systems. In general, this problem is NP-complete and it is proved that for n = 2 and 3 this problem is polynomial-time solvable. In this paper, we prove that for n = 4, the problem is also polynomial-time solvable and give a method to construct the corresponding graph.