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arXiv · 2312.11199

Strong Edge Geodetic Problem on Complete Multipartite Graphs and some Extremal Graphs for the Problem

Sandi Klavžar·Eva Zmazek
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Abstract

A set of vertices $X$ of a graph $G$ is a strong edge geodetic set if to any pair of vertices from $X$ we can assign one (or zero) shortest path between them such that every edge of $G$ is contained in at least one on these paths. The cardinality of a smallest strong edge geodetic set of $G$ is the strong edge geodetic number ${\rm sg_e}(G)$ of $G$. In this paper, the strong edge geodetic number of complete multipartite graphs is determined. Graphs $G$ with ${\rm sg_e}(G) = n(G)$ are characterized and ${\rm sg_e}$ is determined for Cartesian products $P_n\,\square\, K_m$. The latter result in particular corrects an error from the literature.

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Sandi Klavžar, Eva Zmazek. 2023-12-18. Strong Edge Geodetic Problem on Complete Multipartite Graphs and some Extremal Graphs for the Problem. https://arxiv.org/abs/2312.11199

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