WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 13, 2014
nr-Excellent Graphs
Authors: , , ,
Abstract: Given an $$k-$$ tuple of vectors, $$S = (v1, v2, . . . , vk)$$, the neighbourhood adjacency code of a vertex $$v$$ with respect to $$S$$, denoted by $$nc_{S}(v)$$ and defined by $$(a1, a2, . . . , ak)$$ where $$a_{i}$$ is 1 if $$v$$ and $$v_{i}$$ are adjacent and 0 otherwise. $$S$$ S is called a neighbourhood resolving set or a neighbourhood $$r-$$set if $$nc_{S}(u)
eq nc_{S}(v)$$ for any $$u, v ∈ V (G)$$. The least(maximum) cardinality of a minimal neighbourhood resloving set of G is called the neighbourhood(upper neighbourhood) resolving number of G and is denoted by $$nr(G) (NR(G))$$. In this article, we consider the $$nr-$$ excellent graphs. For any Graph G, G is $$nr-$$ excellent if every vertex of G is contained in a minimum neighbourhood resolving set of G. We first prove that the union and join of two given $$nr-$$ excellent graphs is $$nr-$$ excellent under certain conditions. Also we prove that a non $$nr-$$ excellent graph G can be embedded in a $$nr-$$ excellent graph H such that $$nr(H) = nr(G)$$ + number of $$nr-$$ bad vertices of G. Some more results are also discussed.
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Pages: 636-645
WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #62