For every non-complete connected graph with clique number at least 3, the local adjacency metric dimension is at most floor(((ω−2)/(ω−1)) n), confirming the long-open conjecture for the local metric dimension.
On the local metric dimension of $K_4$-free graphs
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Let $G$ be a graph of order $ n(G) $, local metric dimension $ \dim_l(G) $, and clique number $ \omega(G) $. It has been conjectured that if $ n(G) \geq \omega(G) + 1 \geq 4 $, then $ \dim_l(G) \leq \left( \frac{\omega(G) - 2}{\omega(G) - 1} \right) n(G) $. In this paper the conjecture is confirmed for the case $ \omega(G) = 3 $. Consequently, a problem regarding the local metric dimension of planar graphs is also resolved.
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Intertwining local (adjacency) metric dimension with the clique number of a graph
For every non-complete connected graph with clique number at least 3, the local adjacency metric dimension is at most floor(((ω−2)/(ω−1)) n), confirming the long-open conjecture for the local metric dimension.