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Metric dimension related parameters in graphs: A survey on combinatorial, computational and applied results
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Topics concerning metric dimension related invariants in graphs are nowadays intensively studied. This compendium of combinatorial and computational results on this topic is an attempt of surveying those contributions that are of the highest interest for the research community dealing with several variants of metric dimension in graphs.
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Cited by 10 Pith papers
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Mixed metric dimension of $2$-connected graphs
For every 2-connected non-cycle graph G, dim_m(G) ≤ 2c(G), resolving the Sedlar–Škrekovski conjecture.
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On the $(k,\ell)$-multiset anonymity measure for social graphs
The paper defines and analyzes k-multiset antiresolving sets, the basis for a multiset variant of (k, ℓ)-anonymity, and provides an ILP to compute the minimum attacker set size.
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On the local metric dimension of $K_4$-free graphs
Every graph with at least four vertices and no K4 subgraph has local metric dimension at most floor(n/2), confirming the clique-number conjecture when the clique number is 3.
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On the weak $k$-metric dimension of Hamming graphs
The weak k-metric dimension of K_n□K_n is determined exactly for all n≥3 and 2≤k≤2n, complementing the known k=1 case.
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Distance-based (and path-based) covering problems for graphs of given cyclomatic number
For every connected graph, the distance-edge-monitoring number is at most the cyclomatic number plus one, and similar linear bounds hold for metric dimension, geodetic number, and isometric path covers.
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Metric Dimension of a Direct Product of Three Complete Graphs: The Middle Cone Family
For direct products of three complete graphs whose dimensions lie in the middle cone, the metric dimension and location-total-domination number of K(n+1) both equal 2(n3+1)-1.
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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.
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Multiset resolvability parameters in graphs: A survey with new results and open problems
Multiset resolvability parameters are surveyed; sharp outer-multiset lower bounds for diameter-two and join graphs are proved, and block graphs with local multiset dimension two are characterized.
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Counting geodesic paths in graphs
The paper defines the geodesic subpath number and claims an upper bound for it, but the bound is violated by simple graphs such as P3 and K5−e.
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A Survey on Multiset Dimension and Its Variations
A literature survey consolidates results on multiset dimension and its local/outer/edge variants and proposes new multiset partition and related parameters as open directions.
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