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Bar Visibility Numbers for Hypercubes and Outerplanar Digraphs

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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 \(b(G)=1\).

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Authors and Affiliations

  1. Zhejiang Normal University, Jinhua, China

    Douglas B. West

  2. University of Illinois, Urbana, USA

    Douglas B. West

  3. Virginia Polytechnic Institute and State University, Blacksburg, USA

    Jennifer I. Wise

Authors
  1. Douglas B. West
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  2. Jennifer I. Wise
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Corresponding author

Correspondence to Douglas B. West.

Additional information

D. B. West: Research supported by Recruitment Program of Foreign Experts, 1000 Talent Plan, State Admin. of Foreign Experts Affairs, China.

J. I. Wise Research partially supported by NSF Grant DMS 08-38434, “EMSW21-MCTP: Research Experience for Graduate Students”.

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West, D.B., Wise, J.I. Bar Visibility Numbers for Hypercubes and Outerplanar Digraphs. Graphs and Combinatorics 33, 221–231 (2017). https://doi.org/10.1007/s00373-016-1745-4

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  • DOI: https://doi.org/10.1007/s00373-016-1745-4

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