Mutual-Visibility and General Position Sets in Sierpiński Triangle Graphs

Abstract For a given graph G , the general position problem asks for the largest size of a set of vertices $$M \subseteq V(G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>⊆</mml:mo> <mml:mi>V</mml:mi> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> such that no three distinct vertices of M belong to a common shortest path in G . A relaxation of this concept is based on the condition that two vertices $$x, y \in V(G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>V</mml:mi> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> are M -visible, meaning there exists a shortest x , y -path in G that does not pass through any vertex of $$M \setminus \{x, y\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>\</mml:mo> <mml:mo>{</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> . If every pair of vertices in M is M -visible, then M is called a mutual-visibility set of G . The cardinality of the largest mutual-visibility set of G is called the mutual-visibility number of G . Some well-known variations of this concept consider the total, outer, and dual mutual-visibility sets of a graph. We present results on the general position problem and the various mutual-visibility problems in Sierpiński triangle graphs.