# The Connected Edge-To-Vertex Geodetic Number of a Graph

## Authors

• Department of Mathematics, Government College of Engineering, Tirunelveli- 627007 ,IN
• Department of Mathematics, Holy Cross College (Autonomous), Nagercoil ,IN

## Keywords:

Geodesic, Edge-To-Vertex Godetic Number, Connected Edge-To-Vertex Geodetic Number.

## Abstract

Let G = (V, E) be a graph. A subset S ⊆ E is called an edge-to-vertex geodetic set of G if every vertex of G is either incident with an edge of S or lies on a geodesic joining a pair of edges of S. The minimum cardinality of an edge-to-vertex geodetic set of G is gev(G). Any edge-to-vertex geodetic set of cardinality gev(G) is called an edge-to-vertex geodetic basis of G. A connected edge-to-vertex geodetic set of a graph G is an edge-to-vertex geodetic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected edge-to-vertex geodetic set of G is the connected edge-to-vertex geodetic number of G and is denoted by gcev(G). Some general properties satisfied by this concept are studied. The connected graphs G of size q with connected edge-to-vertex geodetic number 2 or q or q − 1 are characterized. It is shown that for any three positive integers q, a and b with 2 ≤ a ≤ b ≤ q, there exists a connected graph G of size q, gev(G) = a and gcev(G) = b.

2023-03-24

## How to Cite

John, J., & S., sujitha. (2023). The Connected Edge-To-Vertex Geodetic Number of a Graph. The Journal of the Indian Mathematical Society, 90(1-2), 1–12. https://doi.org/10.18311/jims/2023/26328

## Section

Articles  Accepted 2022-01-23
Published 2023-03-24

## References

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

F. Buckley, F. Harary and L. V. Quintas, Extremal results on the Geodetic number of a graph, Scientia A 2, (1988) 17–22.

G. Chartrand, F. Harary and P. Zhang, Geodetic sets in Graphs , Discuss. Math. Graph Theory, 20 (2000), 129–138. DOI: https://doi.org/10.7151/dmgt.1112

G. Chartrand, F. Harary, P.Zhang, On the geodetic number of a graph, Networks, 39(1) (2002), 1–6. DOI: https://doi.org/10.1002/net.10007