115 research outputs found
On the Metric Dimension of Cartesian Products of Graphs
A set S of vertices in a graph G resolves G if every vertex is uniquely
determined by its vector of distances to the vertices in S. The metric
dimension of G is the minimum cardinality of a resolving set of G. This paper
studies the metric dimension of cartesian products G*H. We prove that the
metric dimension of G*G is tied in a strong sense to the minimum order of a
so-called doubly resolving set in G. Using bounds on the order of doubly
resolving sets, we establish bounds on G*H for many examples of G and H. One of
our main results is a family of graphs G with bounded metric dimension for
which the metric dimension of G*G is unbounded
Locating and Total Dominating Sets of Direct Products of Complete Graphs
A set S of vertices in a graph G = (V,E) is a metric-locating-total dominating set of G if every vertex of V is adjacent to a vertex in S and for every u ≠ v in V there is a vertex x in S such that d(u,x) ≠ d(v,x). The metric-location-total domination number \gamma^M_t(G) of G is the minimum cardinality of a metric-locating-total dominating set in G. For graphs G and H, the direct product G × H is the graph with vertex set V(G) × V(H) where two vertices (x,y) and (v,w) are adjacent if and only if xv in E(G) and yw in E(H). In this paper, we determine the lower bound of the metric-location-total domination number of the direct products of complete graphs. We also determine some exact values for some direct products of two complete graphs
The Metric Dimension of Graph with Pendant Edges
For an ordered set W = {w_1,w_2,...,w_k} of vertices and a vertex
v in a connected graph G, the representation of v with respect to
W is the ordered k-tuple r(v|W) = (d(v,w_1), d(v,w_2),..., d(v,w_k))
where d(x,y) represents the distance between the vertices x and y.
The set W is called a resolving set for G if every two vertices of G
have distinct representations. A resolving set containing a minimum
number of vertices is called a basis for G. The dimension of G,
denoted by dim(G), is the number of vertices in a basis of G. In this
paper, we determine the dimensions of some corona graphs G⊙K_1,
and G⊙K_m for any graph G and m ≥ 2, and a graph with pendant
edges more general than corona graphs G⊙K_m
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