# The total double geodetic number of a graph

### 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.### References

F. Buckley and F. Harary, Distance in graphs, Redwood City, CA: Addison-Wesley, 1990.

G. Chartrand, F. Harary and P. Zhang, “On the geodetic number of a graph”, Networks, vol. 39, no. 1, pp. 1-6, Nov. 2002, doi: 10.1002/net.10007.

G. Chartrand, F. Harary , H. C. Swart and P. Zhang, “Geodomination in graphs”, Bulletin of the ICA, vol. 31, pp. 51-59, 2001.

F. Harary, Graph theory, Reading, MA: Addision-Wesley, 1969.

F. Harary, E. Loukakis, and C. Tsouros, “The geodetic number of a graph”, Mathematical and computer modeling, vol. 17, no. 11, pp. 89-95, Jun. 1993, doi: 10.1016/0895-7177(93)90259-2.

R. Muntean and P. Zhang, “On geodomonation in graphs”, Congressus numerantium, vol. 143, pp. 161-174, 2000.

P. A. Ostrand, “Graphs with specified radius and diameter”, Discrete mathematics, vol. 4, no. 1, pp. 71-75, 1973, doi: 10.1016/0012-365X(73)90116-7.

A. P. Santhakumaran and T. Jebaraj, “Double geodetic number of a graph”, Discussiones mathematicae graph theory, vol. 32, no. 1, pp. 109-119, 2012. [On line]. Available: https://bit.ly/2Ocj0uW

A. P. Santhakumaran and T. Jebaraj, “The connected double geodetic number of a graph”, (communicated).

*Proyecciones (Antofagasta, On line)*, vol. 39, no. 1, pp. 167-178, Feb. 2020.

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