More on the outer connected geodetic number of a graph.

For a connected graph G = (V , E) of order at least two, a total outer connected geodetic set  S of a graph G is an outer connected geodetic set such that the subgraph induced by S has no isolated vertices. The minimum cardinality of a total outer connected geodetic set of G is the total outer connected geodetic number of G and is denoted by cg to (G). We determine bounds for it and also find the total outer connected geodetic number for some special classes of graphs. It is shown that for positive integers r , d and k ≥ 4 with r < d ≤ 2 r , there exists a connected graph G with rad (G) = r , diam (G) = d and cg to (G) = k. It is proved that for each triple p , d and k of positive integers with k ≥ 4 , d ≥ 2 and p − d − k + 2 ≥ 0 , there exists a connected graph G of order p such that diam (G) = d and cg to (G) = k. It is also shown that for positive integers a , b such that 3 ≤ a ≤ b with b ≤ 2 a , there exists a connected graph G such that g oc (G) = a and cg to (G) = b , where g oc (G) is the outer connected geodetic number of a graph G. [ABSTRACT FROM AUTHOR]