The weighted sitting closer to friends than enemies problem in the line

The weighted Sitting Closer to Friends than Enemies (SCFE) problem is find an injection of the vertex set a given graph into metric space so that, for every pair incident edges with different weight, end vertices heavier edge are closer lighter edge. Seriation simultaneous reordering rows and columns symmetric matrix such that entries monotone nondecreasing in when moving towards diagonal. If exists, it called Robinson ordering. In this work, we establish connection between SCFE problem. We show if extended adjacency G has no ordering then $$\mathbb {R}$$ solves On other hand, ordering, construct polyhedron not empty only there As consequence these results, conclude deciding existence (and constructing) complete can be done polynomial time. incomplete NP-Complete.