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An example of a subgraph G i . The black vertices are included in the set S constructed in the proof of Lemma 11. The cycle with the solid edges is a Hamiltonian cycle of G ′ i , and the dashed lines within the cycle represent edges of G ′ i not in the Hamiltonian cycle. The dotted lines depict a possible partition of the vertices considered in steps 1-3 to obtain an upper bound on |S|.
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We study upper bounds on the size of optimum locating-total dominating sets in graphs. A set $S$ of vertices of a graph $G$ is a locating-total dominating set if every vertex of $G$ has a neighbor in $S$, and if any two vertices outside $S$ have distinct neighborhoods within $S$. The smallest size of such a set is denoted by $\gamma^L_t(G)$. It has...
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