(u, v) of G. 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 (v) for every v ∈ 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.
Consider a problem where we are given a bipartite graph H with vertices arranged on two horizontal lines in the plane, such that the two sets of vertices placed on the two lines form a bipartition of H. We additionally require that H admits a perfect matching and assume that edges of H are embedded in the plane as segments. The goal is to compute the minimal number of intersecting edges in a perfect matching in H.The problem stems from so-called token swapping problems, introduced by Yamanaka et al. [6] and generalized by Bonnet, Miltzow and Rzążewski [1]. We show that our problem, equivalent to one of the special cases of one of the token swapping problems, is NP-complete.
Abstract. 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.
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