A bar layout is a set of vertically oriented non-intersecting line segments, called bars, embedded in the plane. The visibility graph associated with a layout is defined as the graph whose vertices correspond to the bars and whose edges represent the horizontal visibilities between pairs of bars. Characterizations of the bar-representable graphs (those graphs which are the visibility graphs of some bar layout) are known (Tamassia and Tollis, 1986; Wismath, 1985), and directed versions of the problem of determining whether a graph is bar-representable have also been considered (see (S.K. Wismath, 1989)). In this paper, a weighted version of the problem is formulated and solved; namely, given a weighted graph, does there exist a corresponding layout in which the edge weights are commensurate with the amount of visibility between the corresponding pairs of bars. In particular, a polynomial time algorithm for the layout of such graphs is developed when ordering information is specified (for example, when an embedding of the graph must be respected). Determining bar-representability for weighted undirected graphs is equivalent to solving the following flow problem. Given an undirected weighted graph G, attach a source and sink to G, so that for some orientation of the edges, the resulting network is planar, acyclic, and uses all arcs of G to full capacity. We call this the Full Flow Existence problem. It is the connection of this flow problem to the problem of determining bar-representability of graphs that is exploited to solve the weighted case of the latter. Under a different model of visibility (the so-called "strong" model), the characterization results differ markedly, being NP-Complete in the undirected unweighted case (Andreae, 1992). Tamassia and Tollis (1986) presented conditions for the strong visibility representation of directed unweighted ordered graphs were presented. In Section 5, these conditions are strengthened for the weighted case.
Kirkpatrick, D. G., & Wismath, S. K. (1996). Determining bar-representability for ordered weighted graphs. Computational Geometry: Theory and Applications, 6(2), 99–122. https://doi.org/10.1016/0925-7721(95)00017-8