On the edge-to-vertex geodetic number of a graph
A. P. Santhakumaran, J. John · Miskolc mathematical notes/Mathematical notes · 2012
Let G D .V; E/ be a connected graph with at least three vertices.For vertices u andis either incident with an edge of S or lies on a geodesic joining a pair of edges of S: The edge-to-vertex geodetic number g ev .G/ of G is the minimum cardinality of its edge-to-vertex geodetic sets and any edge-to-vertex geodetic set of cardinality g ev .G/ is an edge-to-vertex geodetic basis of G: Any edge-to-vertex geodetic basis is also called aIt is proved that, for a tree T with q 2; g ev .T / D q d C 2 if and only if T is a caterpillar.For positive integers r; d and l 2 with r Ä d Ä 2r; there exists a connected graph G with rad G D r; d iam G D d and g ev .G/ D l: Also graphs G for which g ev .G/ D q; q 1 or q 2 are characterized.