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Open AccessFeature PaperArticle

Secure Resolving Sets in a Graph

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Research Scholar, Register No. 10445, The M.D.T. Hindu College, Tirunelveli 627 010 Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli 627 012, Tamilnadu, India
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Principal, Department of Mathematics, The M.D.T. Hindu College, Tirunelveli 627 010, India
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Author to whom correspondence should be addressed.
Symmetry 2018, 10(10), 439; https://doi.org/10.3390/sym10100439
Received: 5 September 2018 / Revised: 19 September 2018 / Accepted: 19 September 2018 / Published: 27 September 2018
(This article belongs to the Special Issue Symmetry in Graph Theory)
Let G = (V, E) be a simple, finite, and connected graph. A subset S = {u1, u2, , uk} of V(G) is called a resolving set (locating set) if for any xV(G), the code of x with respect to S that is denoted by CS (x), which is defined as CS (x) = (d(u1, x), d(u2, x), .., d(uk, x)), is different for different x. The minimum cardinality of a resolving set is called the dimension of G and is denoted by dim(G). A security concept was introduced in domination. A subset D of V(G) is called a dominating set of G if for any v in V – D, there exists u in D such that u and v are adjacent. A dominating set D is secure if for any u in V – D, there exists v in D such that (D – {v}) ∪ {u} is a dominating set. A resolving set R is secure if for any sV – R, there exists rR such that (R – {r}) ∪ {s} is a resolving set. The secure resolving domination number is defined, and its value is found for several classes of graphs. The characterization of graphs with specific secure resolving domination number is also done. View Full-Text
Keywords: resolving set; domination; secure resolving set and secure resolving domination resolving set; domination; secure resolving set and secure resolving domination
MDPI and ACS Style

Subramanian, H.; Arasappan, S. Secure Resolving Sets in a Graph. Symmetry 2018, 10, 439.

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