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## Print version ISSN 0716-0917

### Proyecciones (Antofagasta) vol.39 no.1 Antofagasta Feb. 2020

#### http://dx.doi.org/10.22199/issn.0717-6279-2020-01-0011

Artículos

The total double geodetic number of a graph

1 Hindustan Institute of Technology and Science, Dept. of Mathematics, Chennai, TN, India. E-mail : apskumar1953@gmail.com

2 Malankara Catholic College, Dept. of Mathematics, Kaliyakavilai, TN, India. E-mail : jebaraj.math@gmail.com

Abstract

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 ∈ S such that x, y ∈ 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.

Keywords: Geodetic number; Double geodetic number; Connected double geodetic number; Total double geodetic number

Acknowledgement.

The authors are thankful to the reviewer for the useful comments for the improvement of this paper.

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Received: January 2019; Accepted: August 2019 This is an open-access article distributed under the terms of the Creative Commons Attribution License