Abstract
Let G be a connected graph and u, v and w vertices of G. Then w is said to resolve u and v if the distance from u to w does not equal the distance from v to w. If there is either a shortest u-w path that contains v or a shortest v-w path that contains u, then we say that w strongly resolves u and v. A set W of vertices of G is a resolving set (strong resolving set), if every pair of vertices of G is resolved (respectively, strongly resolved) by some vertex of W. A smallest resolving set (strong resolving set) of a graph is called a basis (respectively, a strong basis) and its cardinality, denoted β(G) (respectively, β s(G)), the metric dimension (respectively, the strong dimension) of G. The threshold dimension (respectively, threshold strong dimension) of a graph G, denoted τ(G) (respectively, τ s(G)), is the smallest metric dimension (respectively, strong dimension) among all graphs having G as a spanning subgraph. We survey results on the threshold dimension and threshold strong dimension and contrast these invariants. The paper concludes with several open problems for these parameters.
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Notes
- 1.
The interval between two vertices x and y in a graph G is the collection of all vertices that lie on some shortest x–y path.
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Acknowledgements
The work of O. R. Oellermann was partially supported by an NSERC Grant CANADA, Grant number RGPIN-2016-05237.
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Benakli, N., Bong, N.H., Dueck (Gosselin), S., Novick, B., Oellermann, O.R. (2021). The Threshold Dimension and Threshold Strong Dimension of a Graph: A Survey. In: Ferrero, D., Hogben, L., Kingan, S.R., Matthews, G.L. (eds) Research Trends in Graph Theory and Applications. Association for Women in Mathematics Series, vol 25. Springer, Cham. https://doi.org/10.1007/978-3-030-77983-2_4
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