Bar Visibility Numbers for Hypercubes and Outerplanar Digraphs

Abstract

A t-bar visibility representation of a graph G assigns each vertex a union of at most t horizontal segments (“bars”) in the plane so that vertices u and v are adjacent if and only if some bar assigned to u and some bar assigned to v are joined by an unobstructed vertical line of sight having positive width. For an oriented graph G, having an edge uv requires visibility from a bar for u upward to a bar for v. The visibility number of a graph or digraph G, written b(G), is the least t such that G has a t-bar visibility representation. For the n-dimensional hypercube $Q_n$, Euler’s Formula yields $b(Q_n)\ge \lceil \frac{n+1}{4}\rceil$. To prove equality, we decompose $Q_{4k-1}$ explicitly into k spanning subgraphs whose components have the form . When G is an outerplanar oriented graph, always $b(G)\le 2$; we give a forbidden substructure characterization for those with $b(G)=1$.