On the computational complexity of the Steiner $k$-eccentricity

TL;DR

This paper investigates the computational complexity of Steiner $k$-eccentricity, providing efficient algorithms for specific graph classes like block graphs and trees, and analyzing the complexity for general graphs.

Contribution

It introduces new algorithms for computing Steiner $k$-eccentricity efficiently on block graphs and trees, and analyzes the complexity for general graphs.

Findings

Linear time algorithm for block graphs

Improved quadratic algorithm for trees

Complexity analysis for general graphs

Abstract

The Steiner $k$-eccentricity of a vertex $v$ of a graph $G$ is the maximum Steiner distance over all $k$-subsets of $V (G)$ which contain $v$. A linear time algorithm for calculating the Steiner $k$-eccentricity of a vertex on block graphs is presented. For general graphs, an $O(n^{\nu(G)+1}(n(G) + m(G) + k))$ algorithm is designed, where $\nu(G)$ is the cyclomatic number of $G$. A linear algorithm for computing the Steiner $3$-eccentricities of all vertices of a tree is also presented which improves the quadratic algorithm from [Discrete Appl.\ Math.\ 304 (2021) 181--195].