Mutual-visibility in strong products of graphs via total mutual-visibility

Serafino Cicerone, Gabriele Di Stefano, Sandi Klavzar, Ismael G. Yero

Abstract: Let G<math><mi is="true">G</mi></math> be a graph and X⊆V(G)<math><mrow is="true"><mi is="true">X</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">⊆</mo><mi is="true">V</mi><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">)</mo></mrow></mrow></math>. Then X<math><mi is="true">X</mi></math> is a mutual-visibility set if each pair of vertices from X<math><mi is="true">X</mi></math> is connected by a geodesic with no internal vertex in X<math><mi is="true">X</mi></math>. The mutual-visibility number μ(G)<math><mrow is="true"><mi is="true">μ</mi><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">)</mo></mrow></mrow></math> of G<math><mi is="true">G</mi></math> is the cardinality of a largest mutual-visibility set. In this paper, the mutual-visibility number of strong product graphs is investigated. As a tool for this, total mutual-visibility sets are introduced. Along the way, basic properties of such sets are presented. The (total) mutual-visibility number of strong products is bounded from below in two ways, and determined exactly for strong grids of arbitrary dimension. Strong prisms are studied separately and a couple of tight bounds for their mutual-visibility number are given.