The Connected Edge-To-Vertex Geodetic Number of a Graph
Keywords:Geodesic, Edge-To-Vertex Godetic Number, Connected Edge-To-Vertex Geodetic Number.
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.
How to Cite
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
F. Harary, Graph Theory, Addison-Wesley, 1969. DOI: https://doi.org/10.21236/AD0705364
D. A. Mojdeh and N. J. Rad, Connected geodomination in graphs, J. Discrete Math. Sci. Cryptogr., 9(1) (2006), 177–186. DOI: https://doi.org/10.1080/09720529.2006.10698070
A. P. Santhakumaran P. Titus and J. John, On the connected geodetic number of a Graph, J. Comb. Maths. Comb. Comput, 69 (2009), 219–229.
A. P. Santhakumaran P. Titus and J. John, The upper connected geodetic number and the forcing connected geodetic number of a Graph, Discrete Appl. Math., 157(7) (2009), 1571–1580. DOI: https://doi.org/10.1016/j.dam.2008.06.005
A. P. Santhakumaran and J. John, On the edge-to-vertex geodetic number of a graph, Miskolc Math. Notes, 13(1) (2012), 131-141. DOI: https://doi.org/10.2298/FIL1201131S