GRAPHS WITH MINIMUM OR MAXIMUM LOCAL METRIC DIMENSION
Let W = fw1;w2; :::;wkg be an ordered set consisting of k distinct vertices in a nontrivial connected graph G. The metric representation of a vertex v in G with respect to W is the k-vector r(vjW) = (d(v;w1); d(v;w2); :::; d(v;wk)) where d(v;wi) represents the distance between v and wi for some...
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| Format: | Final Project |
| Language: | Indonesia |
| Online Access: | https://digilib.itb.ac.id/gdl/view/42078 |
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| Institution: | Institut Teknologi Bandung |
| Language: | Indonesia |
| Summary: | Let W = fw1;w2; :::;wkg be an ordered set consisting of k distinct vertices in a
nontrivial connected graph G. The metric representation of a vertex v in G with
respect to W is the k-vector
r(vjW) = (d(v;w1); d(v;w2); :::; d(v;wk))
where d(v;wi) represents the distance between v and wi for some 1 i k. If for
any arbitrary pair of adjacent vertices u and v in G satisfies r(ujW) 6= r(vjW), then
W is a local metric set of G. The smallest positive integer k such that G has a local
metric k-set is called the local metric dimension of G (lmd(G)). A local metric set
of G having cardinality lmd(G) is a local metric basis of G.
Moreover, let n be the order of G, l be the number of true twin equivalence classes
in G, and d be the diameter of G. Then, G satisfies lmd(G) n????l and lmd(G)
n ???? d:
In this final project, some nontrivial connected graphs G with lmd(G) = n ???? l or
lmd(G) = n ???? d will be presented. |
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