The Partial Visibility Representation Extension Problem

For a graph G, a function ψ is called a bar visibility representation of G when for each vertex v∈V(G), ψ(v) is a horizontal line segment (bar) and uv∈E(G) iff there is an unobstructed, vertical, ε-wide line of sight between ψ(u) and ψ(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 ψ of G, additionally, for each directed edge (u, v) of G, puts the bar ψ(u) strictly below the bar ψ(v). We study a generalization of the recognition problem where a function ψ′ defined on a subset V′ of V(G) is given and the question is whether there is a bar visibility representation ψ of G with ψ|V′=ψ′. We show that for undirected graphs this problem together with closely related problems are NP -complete, but for certain cases involving directed graphs it is solvable in polynomial time