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Net."],"published-print":{"date-parts":[[2025,6]]},"abstract":"<jats:p> Let [Formula: see text] be a graph and [Formula: see text] be the vertex set of [Formula: see text]. If any edge appears in the unique shortest path of at least one vertex pair in set [Formula: see text], then [Formula: see text] is defined as the monitoring edge-geodetic set (MEG-set for short). We denote by [Formula: see text] the size of a smallest MEG-set of [Formula: see text]. 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