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The partition dimension for a subdivision of homogeneous caterpillars
Amrullaha, Assiyatun H.a, Baskoro E.T.a, Uttunggadewa S.a, Simanjuntak R.a
a Combinatorial Mathematics Research Group, Institut Teknologi Bandung (ITB), Indonesia
[vc_row][vc_column][vc_row_inner][vc_column_inner][vc_separator css=”.vc_custom_1624529070653{padding-top: 30px !important;padding-bottom: 30px !important;}”][/vc_column_inner][/vc_row_inner][vc_row_inner layout=”boxed”][vc_column_inner width=”3/4″ css=”.vc_custom_1624695412187{border-right-width: 1px !important;border-right-color: #dddddd !important;border-right-style: solid !important;border-radius: 1px !important;}”][vc_empty_space][megatron_heading title=”Abstract” size=”size-sm” text_align=”text-left”][vc_column_text]The concept of graph partition dimension was introduced by Chartrand et al. (1998). Let G = (V,E) be a connected graph. For every v ∈ V(G) and L ⊆ V(G), define the distance from v to L as d(v, L) = min{d(v, w){pipe}w ∈ L}. Let P{cyrillic} = {L1,L2, · · ·,Lk} be a partition of V(G). The representation of a vertex v with respect to P{cyrillic} is defined as r(v{pipe}P{cyrillic}) = (d(v,L1), d(v,L2), · · ·, d(v, Lk)). The partition P{cyrillic} is called a resolving partition of G if all representations of the vertices are distinct. The partition dimension of a graph G can be defined as the cardinality of a minimum resolving partition P{cyrillic} of G. Let e ∈ V (G) and k ≥ 1. The subdivision S(G(e; k)) of a graph G on e is a graph obtained from graph G by replacing edge e with a path on k + 2 vertices. In this paper, we determine the partition dimension of S(G(e; k)) with G ≃ C(m; r) is a homogeneous caterpillar. We show that pd(S(G(e; k))) = pd(G) for almost all values of m, r and k.[/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”Author keywords” size=”size-sm” text_align=”text-left”][vc_column_text][/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”Indexed keywords” size=”size-sm” text_align=”text-left”][vc_column_text]Caterpillar,Homogeneous,Partition dimension,Resolving partition,Subdivision[/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”Funding details” size=”size-sm” text_align=”text-left”][vc_column_text][/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”DOI” size=”size-sm” text_align=”text-left”][vc_column_text][/vc_column_text][/vc_column_inner][vc_column_inner width=”1/4″][vc_column_text]Widget Plumx[/vc_column_text][/vc_column_inner][/vc_row_inner][/vc_column][/vc_row][vc_row][vc_column][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][/vc_column][/vc_row]