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arXiv · 2306.00843

Separating path systems in trees

Francisco Arrepol·Patricio Asenjo·Raúl Astete·Víctor Cartes·Anahí Gajardo·Valeria Henríquez·Catalina Opazo·Nicolás Sanhueza-Matamala·Christopher Thraves Caro
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Abstract

For a graph $G$, an edge-separating (resp. vertex-separating) path system of $G$ is a family of paths in $G$ such that for any pair of edges $e_1, e_2$ (resp. pair of vertices $v_1, v_2$) of $G$ there is at least one path in the family that contains one of $e_1$ and $e_2$ (resp. $v_1$ and $v_2$) but not the other. We determine the size of a minimum edge-separating path system of an arbitrary tree $T$ as a function of its number of leaves and degree-two vertices. We obtain bounds for the size of a minimal vertex-separating path system for trees, which we show to be tight in many cases. We obtain similar results for a variation of the definition, where we require the path system to separate edges and vertices simultaneously. Finally, we investigate the size of a minimal vertex-separating path system in Erd\H{o}s--R\'enyi random graphs.

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Francisco Arrepol, Patricio Asenjo, Raúl Astete, Víctor Cartes, Anahí Gajardo, Valeria Henríquez, Catalina Opazo, Nicolás Sanhueza-Matamala, Christopher Thraves Caro. 2023-06-01. Separating path systems in trees. https://arxiv.org/abs/2306.00843

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