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New classes of convex polytopes with constant metric dimension
Imran M.a, Baig A.Q.b, Bokhary S.A.c, Baskoro E.T.a,d
a Center for Advanced Mathematics and Physics (CAMP), National University of Science and Technology (NUST), Islamabad, Pakistan
b Department of Mathematics, GC University Faisalabad, Faisalabad, Pakistan
c Center for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan
d 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]A family G of connected graphs is a family with constant metric dimension if dim(G) is finite and does not depend upon the choice of G in G. The metric dimension of some classes of plane graphs has been determined in [2], [3], [4], [12], [15] and [22] while metric dimension of some families of convex polytopes has been studied in [9], [10] and [11] and the following open problem was raised in [10]. Open Problem [10]: Let G’ be the graph of convex polytope obtained from the graph of convex polytope G by adding new edges in G such that V(G’) = V(G). Is it the case that G’ and G will always have the same metric dimension? In this paper, we extend this study by considering some classes of convex polytopes which are obtained from the graph of convex polytope Sn defined in [11] by adding new edges in it and having the same vertex set. It is shown that these classes of convex polytoes have constant metric dimension and only three vertices chosen appropriately suffice to resolve all the vertices of these classes of convex polytopes. A conjecture in more general meaning is also proposed in this regard. It is natural to ask for the characterization of classes of convex polytopes with constant metric dimension.[/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]Basis,Convex polytope,Metric dimension,Plane graph,Resolving set[/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]