@article{Santhakumaran_Jebaraj_2020, title={The total double geodetic number of a graph}, volume={39}, url={https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/3984}, DOI={10.22199/issn.0717-6279-2020-01-0011}, abstractNote={<p><em>For a connected graph G of order n, a set S of vertices is called a double geodetic set of G if for each pair of vertices x, y in G there exist vertices u, v </em><em>∈</em><em> S such that x, y </em><em>∈</em><em> I[u, v]. The double geodetic number dg(G) is the minimum cardinality of a double geodetic set. Any double godetic set of cardinality dg(G) is called a dg-set of G. A connected double geodetic set of G is a double geodetic set S such that the subgraph G[S] induced by S is connected. The mínimum cardinality of a connected double geodetic set of G is the connected double geodetic number of G and is denoted by dgc(G). A connected double geodetic set of cardinality dgc(G) is called a dgc-set of G. A total double geodetic set of a graph G is a double geodetic set S such that the subgraph G[S] induced by S has no isolated vertices. The minimum cardinality of a total double geodetic set of G is the total double geodetic number of G and is denoted by dgt(G). For positive integers r, d and k ≥ 4 with r ≤ d ≤ 2r, there exists a connected graph G with rad G = r, diam G = d and dgt(G) = k. It is shown that if n, a, b are positive integers such that 4 ≤ a ≤ b ≤ n, then there exists a connected graph G of order n with dgt(G) = a and dgc(G) = b. Also, for integers a, b with 4 ≤ a ≤ b and b ≤ 2a, there exists a connected graph G such that dg(G) = a and dgt(G) = b.</em></p>}, number={1}, journal={Proyecciones (Antofagasta, On line)}, author={Santhakumaran, A. P. and Jebaraj, T.}, year={2020}, month={Feb.}, pages={167-178} }