162 research outputs found
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 β v in V there is a vertex x in S such that d(u,x) β 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 Γ H is the graph with vertex set V(G) Γ 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 Amalgamation of Cycles
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 vertex of G has a distinct representation. 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. Let {G_i} be a finite collection of graphs and each G_i has a fixed vertex voi called a terminal. The amalgamation Amal {Gi , v_{oi}} is formed by taking all of the G_iβs and identifying their terminals. In this paper, we determine the metric dimension of amalgamation of cycles
DIMENSI METRIK PADA GRAF TURAN
Himpunan adalah resolving set (himpunan pembeda) dari jika setiap simpul di memiliki representasi tunggal pada yang ditentukan oleh jarak dari simpul dari terhadap simpul di . Dimensi metrik dari adalah kardinalitas minimum dari resolving set pada . Pada paper ini akan dijelaskan tentang dimensi metrik pada graf Turan, yaitu graf multipartisi komplit yang dinotasikan dengan dengan adalah banyaknya seluruh simpul dari graf dan adalah banyaknya partisi. Dimensi metrik dari graf dengan adalah . Sedangkan untuk graf Turan yang tidak komplit yang memiliki order dengan menjaga keterhubungan antara setiap simpul pada semua partisi memiliki dimensi metrik yaitu .
Kata Kunci : dimensi metrik, graf Turan, resolving set
 
Metric Dimension of Amalgamation of Graphs
A set of vertices resolves a graph if every vertex is uniquely
determined by its vector of distances to the vertices in . The metric
dimension of is the minimum cardinality of a resolving set of .
Let be a finite collection of graphs and each
has a fixed vertex or a fixed edge called a terminal
vertex or edge, respectively. The \emph{vertex-amalgamation} of , denoted by , is formed by taking all
the 's and identifying their terminal vertices. Similarly, the
\emph{edge-amalgamation} of , denoted by
, is formed by taking all the 's and identifying
their terminal edges.
Here we study the metric dimensions of vertex-amalgamation and
edge-amalgamation for finite collection of arbitrary graphs. We give lower and
upper bounds for the dimensions, show that the bounds are tight, and construct
infinitely many graphs for each possible value between the bounds.Comment: 9 pages, 2 figures, Seventh Czech-Slovak International Symposium on
Graph Theory, Combinatorics, Algorithms and Applications (CSGT2013), revised
version 21 December 201
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