Abstract
For any two vertices x and y of a graph G, let S{x, y} denote the set of vertices z such that either x lies on a y − z geodesic or y lies on a x − z geodesic. For a function g defined on V(G) and U ⊆ V(G), let g(U) = ∑ x ∈ Ug(x). A function g: V(G) → [0,1] is a strong resolving function of G if g(S{x, y}) ≥ 1, for every pair of distinct vertices x, y of G. The fractional strong metric dimension, sdim f (G), of a graph G is min {g(V(G)): g is a strong resolving function of G}. For any connected graph G of order n ≥ 2, we prove the sharp bounds \(1 \le sdim_f(G) \le \frac{n}{2}\). Indeed, we show that sdim f (G) = 1 if and only if G is a path. If G contains a cut-vertex, then \(sdim_f(G) \le \frac{n-1}{2}\) is the sharp bound. We determine sdim f (G) when G is a tree, a cycle, a wheel, a complete k-partite graph, or the Petersen graph. For any tree T, we prove the sharp inequality sdim f (T + e) ≥ sdim f (T) and show that sdim f (G + e) − sdim f (G) can be arbitrarily large. Lastly, we furnish a Nordhaus-Gaddum-type result: Let G and \(\overline{G}\) (the complement of G) both be connected graphs of order n ≥ 4; it is readily seen that \(sdim_f(G)+sdim_f(\overline{G})=2\) if and only if n = 4; further, we characterize unicyclic graphs G attaining \(sdim_f(G)+sdim_f(\overline{G})=n\).
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References
Arumugam, S., Mathew, V.: The fractional metric dimension of graphs. Discrete Math. 312, 1584–1590 (2012)
Chartrand, G., Eroh, E., Johnson, M.A., Oellermann, O.R.: Resolvability in graphs and the metric dimension of a graph. Discrete Appl. Math. 105, 99–113 (2000)
Currie, J., Oellermann, O.R.: The metric dimension and metric independence of a graph. J. Combin. Math. Combin. Comput. 39, 157–167 (2001)
Fehr, M., Gosselin, S., Oellermann, O.R.: The metric dimension of Cayley digraphs. Discrete Math. 306, 31–41 (2006)
Garey, M.R., Johnson, D.S.: Computers and intractability: A guide to the theory of NP-completeness. Freeman, New York (1979)
Harary, F., Melter, R.A.: On the metric dimension of a graph. Ars Combin. 2, 191–195 (1976)
Holton, D.A., Sheehan, J.: The Petersen graph. Cambridge University Press (1993)
Khuller, S., Raghavachari, B., Rosenfeld, A.: Landmarks in graphs. Discrete Appl. Math. 70, 217–229 (1996)
Nordhaus, E.A., Gaddum, J.W.: On complementary graphs. Amer. Math. Monthly 63, 175–177 (1956)
Oellermann, O.R., Peters-Fransen, J.: The strong metric dimension of graphs and digraphs. Discrete Appl. Math. 155, 356–364 (2007)
Scheinerman, E.R., Ullman, D.H.: Fractal graph theory: A rational approach to the theory of graphs. John Wiley & Sons, New York (1997)
Sebö, A., Tannier, E.: On metric generators of graphs. Math. Oper. Res. 29, 383–393 (2004)
Slater, P.J.: Leaves of trees. Congress. Numer. 14, 549–559 (1975)
Yi, E.: The fractional metric dimension of permutation graphs (submitted)
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Kang, C.X., Yi, E. (2013). The Fractional Strong Metric Dimension of Graphs. In: Widmayer, P., Xu, Y., Zhu, B. (eds) Combinatorial Optimization and Applications. COCOA 2013. Lecture Notes in Computer Science, vol 8287. Springer, Cham. https://doi.org/10.1007/978-3-319-03780-6_8
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DOI: https://doi.org/10.1007/978-3-319-03780-6_8
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