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Fixing Numbers of Point-Block Incidence Graphs

Author

Listed:
  • Josephine Brooks

    (Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada)

  • Alvaro Carbonero

    (Department of Mathematical Sciences, University of Nevada, Las Vegas, NV 89154, USA)

  • Joseph Vargas

    (Mathematical Sciences Department, State University of New York, Fredonia, NY 14063, USA)

  • Rigoberto Flórez

    (Department of Mathematical Sciences, The Citadel, Charleston, SC 29409, USA)

  • Brendan Rooney

    (School of Mathematical and Statistical Sciences, Rochester Institute of Technology, Rochester, NY 14623, USA)

  • Darren A. Narayan

    (School of Mathematical and Statistical Sciences, Rochester Institute of Technology, Rochester, NY 14623, USA)

Abstract

A vertex in a graph is referred to as fixed if it is mapped to itself under every automorphism of the vertices. The fixing number of a graph is the minimum number of vertices, when fixed, that fixes all of the vertices in the graph. Fixing numbers were first introduced by Laison and Gibbons, and independently by Erwin and Harary. Fixing numbers have also been referred to as determining numbers by Boutin. The main motivation is to remove all symmetries from a graph. A very simple application is in the creation of QR codes where the symbols must be fixed against any rotation. We determine the fixing number for several families of graphs, including those arising from combinatorial block designs. We also present several infinite families of graphs with an even stronger condition, where fixing any vertex in a graph fixes every vertex.

Suggested Citation

  • Josephine Brooks & Alvaro Carbonero & Joseph Vargas & Rigoberto Flórez & Brendan Rooney & Darren A. Narayan, 2023. "Fixing Numbers of Point-Block Incidence Graphs," Mathematics, MDPI, vol. 11(6), pages 1-9, March.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:6:p:1322-:d:1092183
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    Keywords

    fixing number; graph automorphism;

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