Abstract
A set S of vertices of a graph G is a geodesic transversal of G if every maximal geodesic of G contains at least one vertex of S. The minimum cardinality of a geodesic transversal of G is denoted by \(\text{ gt }(G)\) and is called geodesic transversal number. For two graphs G and H we deal with the behavior of this invariant for the lexicographic product \(G\circ H\) and join \(G\oplus H\). We determine \(\text{ gt }(G\oplus H)\) in terms of structural properties of the original graphs and describe \(\text{ gt }(G\circ H)\) as a solution of an optimization problem concerning specific subsets of V(G).
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Acknowledgements
The research of the first author was partially supported by the Slovenian research agency by the Grants P1-0297, J1-1693 and J1-9109. The research of the second author was partially supported by the Slovak Grant Agency for Science (VEGA) under contract 1/0177/21. This work was done during the visit of the first author at the Pavol Jozef Šafárik University in Košice, Slovakia using infrastructure developed by the project UVP Technicom ITMS 313011D232.
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Communicated by Leonardo de Lima.
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Peterin, I., Semanišin, G. Geodesic transversal problem for join and lexicographic product of graphs. Comp. Appl. Math. 41, 128 (2022). https://doi.org/10.1007/s40314-022-01834-1
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DOI: https://doi.org/10.1007/s40314-022-01834-1