The Partial Visibility Representation Extension Problem

For a graph G, a function \(\psi \) is called a bar visibility representation of G when for each vertex \(v \in V(G)\), \(\psi (v)\) is a horizontal line segment (bar) and \(uv \in E(G)\) iff there is an unobstructed, vertical, \(\varepsilon \)-wide line of sight between \(\psi (u)\) and \(\psi (v)\). Graphs admitting such representations are well understood (via simple characterizations) and recognizable in linear time. For a directed graph G, a bar visibility representation \(\psi \) of G, additionally, for each directed edge (u, v) of G, puts the bar \(\psi (u)\) strictly below the bar \(\psi (v)\). We study a generalization of the recognition problem where a function \(\psi '\) defined on a subset \(V'\) of V(G) is given and the question is whether there is a bar visibility representation \(\psi \) of G with \(\psi |V' = \psi '\). We show that for undirected graphs this problem together with closely related problems are \(\mathsf {NP}\)-complete, but for certain cases involving directed graphs it is solvable in polynomial time.