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The Partition Dimension of a Subdivision of a Complete Graph
Amrullaha, Baskoro E.T.a, Simanjuntak R.a, Uttunggadewa S.a
a Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Bandung, 40132, 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]© 2015 The Authors.The concept of a graph partition dimension was introduced by Chartrand et al. (1998). Let Π = {L1, L2, L3, ⋯, Lk } be a k-partition of V(G). The representation r(v|Π) of a vertex v with respect to Π is the vector (d(v, L1), d(v, L2), ⋯, d(v, Lk)). The partition Π is called a resolving partition of G if r(w|Π) ≠ r(v|Π) for all distinct w, v ε V(G). The partition dimension of a graph, denoted by pd(G), is the cardinality of a minimum resolving partition of G. This paper considers in finding partition dimensions of graphs obtained from a subdivision operation. In particular, we derive an upper bound of partition dimension of a subdivision of a complete graph Kn with n ≥ 9. Additionally for n ε [2,8], we obtain the exact values of the partition dimensions.[/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]Cardinalities,Complete graphs,Graph partition,K-partition,Partition dimensions,Resolving partitions,subdivision,Upper Bound[/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]complete graph,Partition dimension,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]Acknowledgment. This research was supported by Research Grant ”Riset Unggulan Perguruan Tinggi ITB-DIKTI”, Indonesian Ministry of Research, Technology, and Higher Education.[/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]https://doi.org/10.1016/j.procs.2015.12.075[/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]