Steiner Distance in Graphs--A Survey

TL;DR

This survey reviews the current state of research on Steiner distance parameters in graphs, including definitions, properties, and open problems, providing a comprehensive overview for future investigations.

Contribution

It compiles and summarizes known results on Steiner distance parameters, highlighting conjectures and open problems for further research.

Findings

Summarized key results on Steiner distance and related parameters.

Identified open problems and conjectures in the field.

Provided a comprehensive overview for future research directions.

Abstract

For a connected graph $G$ of order at least $2$ and $S\subseteq V(G)$, the \emph{Steiner distance} $d_G(S)$ among the vertices of $S$ is the minimum size among all connected subgraphs whose vertex sets contain $S$. In this paper, we summarize the known results on the Steiner distance parameters, including Steiner distance, Steiner diameter, Steiner center, Steiner median, Steiner interval, Steiner distance hereditary graph, Steiner distance stable graph, average Steiner distance, and Steiner Wiener index. It also contains some conjectures and open problems for further studies.