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 is an element of V (G), psi (v) is a horizontal line segment (bar) and uv is an element of E (G) if and only if there is an unobstructed, vertical, epsilon-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 of G, additionally, puts the bar psi (u) strictly below the bar psi (v) for each directed edge (u, v) of G. 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. of G with psi(v) = psi' (v) for every v is an element of V'. We show that for undirected graphs this problem, and other closely related problems, is NP-complete, but for certain cases involving directed graphs it is solvable in polynomial time.