The upper connected edge geodetic number of a graph

Abstract

For a non-trivial connected graph G, a set S ⊆ V (G) is called an edge geodetic set of G if every edge of G is contained in a geodesic joining some pair of vertices in S. The edge geodetic number g1(G) of G is the minimum order of its edge geodetic sets and any edge geodetic set of order g1(G) is an edge geodetic basis. A connected edge geodetic set of G is an edge geodetic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected edge geodetic set of G is the connected edge geodetic number of G and is denoted by g1c(G). A connected edge geodetic set of cardinality g1c(G) is called a g1c-set of G or connected edge geodetic basis of G. A connected edge geodetic set S in a connected graph G is called a minimal connected edge geodetic set if no proper subset of S is a connected edge geodetic set of G. The upper connected edge geodetic number g + 1c (G) is the maximum cardinality of a minimal connected edge geodetic set of G. Graphs G of order p for which g1c(G) = g + 1c = p are characterized. For positive integers r,d and n ≥ d + 1 with r ≤ d ≤ 2r, there exists a connected graph of radius r, diameter d and upper connected edge geodetic number n. It is shown for any positive integers 2 ≤ a < b ≤ c, there exists a connected graph G such that g1(G) = a, g1c(G) = b and g + 1c (G) = c.

The edge geodetic self decomposition number of a graph

“…The edge geodetic number g e (G) of G is the minimum order of its edge geodetic sets and any edge geodetic set of order g e (G) is an edge geodetic basis of G or g e -set of G. The edge geodetic number of a graph was introduced by M. Atici [2] and further studied in [1,6,[10][11][12][13][14]. Recently edge geodetic concepts were studied in [9,[15][16][17].…”

The edge-to-edge geodetic domination number of a graph

“…Thus every vertex of G is contained in a geodesic joining some pair of vertices in S. The minimum cardinality of a geodetic set of G is called the geodetic number of G and is denoted by g(G). A geodetic set of minimum cardinality is called g-set of G. For references on geodetic parameters in graphs see [4,5,6,7,10,14,15,22,23].…”

Strong Geodetic Number of Complete Bipartite Graphs and of Graphs with Specified Diameter

“…Connection between geodetic number and diameter of a graph has been studied in [4]. See [1] for characterization of graphs with large geodetic number and [19,20] for results on the edge geodetic problem. For additional results see [5,6] and [21].…”