ON THE LOCAL METRIC DIMENSION OF LINE GRAPHS
Let G = (V;E) be a connected graph. The distance between two vertices u and v in G, denoted by dG(u; v), is the length of a shortest u ???? v path. Suppose W = fw1;w2; : : : ;wkg be an ordered nonempty subset of vertices of G and let v is a vertex in G. The representation of vertex v with respect...
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| Format: | Theses |
| Language: | Indonesia |
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| Online Access: | https://digilib.itb.ac.id/gdl/view/34813 |
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| Institution: | Institut Teknologi Bandung |
| Language: | Indonesia |
| Summary: | Let G = (V;E) be a connected graph. The distance between two vertices u and
v in G, denoted by dG(u; v), is the length of a shortest u ???? v path. Suppose W =
fw1;w2; : : : ;wkg be an ordered nonempty subset of vertices of G and let v is a
vertex in G. The representation of vertex v with respect to W, denoted by r(vjW),
is the k-vector (dG(v;w1); dG(v;w2); : : : ; dG(v;wk)). If every two adjacent vertices
of G has distinct representations with respect to W, then the set W is called a local
resolving set for G. A local resolving set of G with the smallest cardinality is called
a local metric basis for G. The local metric dimension of G is the cardinality of a
local metric basis of G, and denoted by lmd(G).
In this research, we obtained lower and upper bounds of local metric dimension
of line graphs, characterization of connected graphs whose line graphs have local
metric dimension 1, and local metric dimension of line graphs of star graphs,
friendship graphs, fan graphs, wheel graphs, caterpillar graphs, and rooted product
graphs. |
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