A Steiner general position problem in graph theory

TL;DR

This paper introduces the concept of $k$-Steiner general position sets in graphs, explores their properties, and determines their maximum size for various graph classes, providing bounds and exact formulas.

Contribution

It defines $k$-Steiner general position sets, introduces Steiner cliques, and determines the $k$-Steiner general position number for several graph families, including trees, cycles, and lexicographic products.

Findings

Determined ${ m sgp}_k(G)$ for trees, cycles, and joins.

Bounded ${ m sgp}_k(G)$ using Steiner cliques.

Derived an exact formula for lexicographic products.

Abstract

Let $G$ be a graph. The Steiner distance of $W\subseteq V(G)$ is the minimum size of a connected subgraph of $G$ containing $W$. Such a subgraph is necessarily a tree called a Steiner $W$-tree. The set $A\subseteq V(G)$ is a $k$-Steiner general position set if $V(T_B)\cap A = B$ holds for every set $B\subseteq A$ of cardinality $k$, and for every Steiner $B$-tree $T_B$. The $k$-Steiner general position number ${\rm sgp}_k(G)$ of $G$ is the cardinality of a largest $k$-Steiner general position set in $G$. Steiner cliques are introduced and used to bound ${\rm sgp}_k(G)$ from below. The $k$-Steiner general position number is determined for trees, cycles and joins of graphs. Lower bounds are presented for split graphs, infinite grids and lexicographic products. The lower bound for the latter products leads to an exact formula for the general position number of an arbitrary lexicographic product.