Analogies between the geodetic number and the Steiner number of some classes of graphs

TL;DR

This paper explores the relationship between the geodetic and Steiner numbers in corona product graphs, establishing inequalities and characterizing graph families where the geodetic number is less than or equal to the Steiner number.

Contribution

It provides new bounds and partial solutions to an open problem regarding the comparison of geodetic and Steiner numbers in specific graph classes.

Findings

If G is connected with order n≥2 and H is non-complete, then g(G⊙H) ≤ s(G⊙H).

The study partially characterizes graphs where the geodetic number is less than or equal to the Steiner number.

Addresses an open problem from previous literature on graph parameters.

Abstract

A set of vertices $S$ of a graph $G$ is a geodetic set of $G$ if every vertex $v\not\in S$ lies on a shortest path between two vertices of $S$. The minimum cardinality of a geodetic set of $G$ is the geodetic number of $G$ and it is denoted by $g(G)$. A Steiner set of $G$ is a set of vertices $W$ of $G$ such that every vertex of $G$ belongs to the set of vertices of a connected subgraph of minimum size containing the vertices of $W$. The minimum cardinality of a Steiner set of $G$ is the Steiner number of $G$ and it is denoted by $s(G)$. Let $G$ and $H$ be two graphs and let $n$ be the order of $G$. The corona product $G\odot H$ is defined as the graph obtained from $G$ and $H$ by taking one copy of $G$ and $n$ copies of $H$ and joining by an edge each vertex from the $i^{th}$-copy of $H$ with the $i^{th}$-vertex of $G$. We study the geodetic number and the Steiner number of corona product graphs. We show that if $G$ is a connected graph of order $n\ge 2$ and $H$ is a non complete graph, then $g(G\odot H)\le s(G\odot H)$, which partially solve the open problem presented in [\emph{Discrete Mathematics} \textbf{280} (2004) 259--263] related to characterize families of graphs $G$ satisfying that $g(G)\le s(G)$.