A set S of vertices of a graph G is a dominating set of G if every vertex u of G is either in S or it has a neighbour in S. In other words S is dominating if the sets S ∩ N [u] where u ∈ V (G) andwhere u ∈ V (G) are all nonempty and distinct. We study locating and identifying codes in the circulant networks C n (1, 3). For an integer n 7, the graph C n (1, 3) has vertex set Z n and edges xy where x, y ∈ Z n and |x − y| ∈ {1, 3}. We prove that a smallest locating code in C n (1, 3) has size n/3 + c, where c ∈ {0, 1}, and a smallest identifying code in C n (1, 3) has size 4n/11 + c , where c ∈ {0, 1}.
In the paper we present lower bounds for the connectivity of a path graph P 2 (G) of a graph G. Let δ ≥ 3 be the minimum degree of G. We prove that if G is a connected graph, then P 2 (G) is at least (δ−1)connected; and if G is 2-connected, then P 2 (G) is at least (2δ−2)connected. We remark that if G is a δ-regular graph then P 2 (G) is (2δ−2)-regular, and hence, if G is 2-connected then P 2 (G) is (2δ−2)connected; its theoretical maximum.
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