Unique Metro Domination of a Ladder

A dominating set D of a graph G which is also a resolving set of G is called a metro dominating set. A metro dominating set D of a graph G(V,E) is a unique metro dominating set (in short an UMD-set) if |N(v) \cap D| = 1 for each vertex v in V-D and the minimum cardinality of an UMD-set of G is the unique metro domination number of G. In this paper, we determine unique metro domination number of P_n\times P_2

Graphs of Neighborhood Metric Dimension Two

A subset  of vertices of a simple connected graph is a neighborhood set (n-set) of  G if G is the union of subgraphs of G induced by the closed neighbors of elements in S. Further, a set S is a resolving set of G if for each pair of distinct vertices x,y of G, there is a vertex s∈ S such that d(s,x)≠d(s,y). An n-set that serves as a resolving set for G is called an nr-set of G. The nr-set with least cardinality is called an nr-metric basis of G and its cardinality is called the neighborhood metric dimension of graph G. In this paper, we characterize graphs of neighborhood metric dimension two

Open Neighborhood Coloring of Prisms

For a simple, connected, undirected graph G(V, E) an open neighborhood coloring of the graph G is a mapping f : V (G) --> Z+ such that for each w in V(G), and for all u, v in N(w), f(u) is different from f(v). The maximum value of f(w), for all w in V (G) is called the span of the open neighborhood coloring f. The minimum value of span of f over all open neighborhood colorings f is called open neighborhood chromatic number of G, denoted by Xonc(G). In this paper we determine the open neighborhood chromatic number of prisms

Graphs of Neighborhood Metric Dimension Two

A subset  of vertices of a simple connected graph is a neighborhood set (n-set) of  G if G is the union of subgraphs of G induced by the closed neighbors of elements in S. Further, a set S is a resolving set of G if for each pair of distinct vertices x,y of G, there is a vertex s∈ S such that d(s,x)≠d(s,y). An n-set that serves as a resolving set for G is called an nr-set of G. The nr-set with least cardinality is called an nr-metric basis of G and its cardinality is called the neighborhood metric dimension of graph G. In this paper, we characterize graphs of neighborhood metric dimension two

Certain Varieties of Resolving Sets of A Graph

Let G=(V,E) be a simple connected graph. For each ordered subset S={s_1,s_2,...,s_k} of V and a vertex u in V, we associate a vector Gamma(u/S)=(d(u,s_1),d(u,s_2),...,d(u,s_k)) with respect to S, where d(u,v) denote the distance between u and v in G. A subset S is said to be resolving set of G if Gamma(u/S) not equal to Gamma(v/S) for all u, v in V-S. The purpose of this paper is to introduce various types of r-sets and compute minimum cardinality of each set, in possible cases, particulary for paths, cycles, complete graphs and wheels