• THE TOTAL x-EDGE STEINER NUMBER OF A GRAPH

Abstract

For a vertex x of a connected graph G = (V, E) and WV (G) is called total x-edge Steiner set if the subgraph < W > induced by W has no isolated vertex. The minimum cardinality of a total x-edge Steiner set of G is the total x-edge Steiner number of G and denoted by  st1x(G). Some general properties satisfied by this concept are studied. The total  x-edge Steiner number of certain classes of graphs are determined.  Necessary conditions for connected graph of order p with total x-edge Steiner number to be p−1 is given. It is shown that for positive integers r, d and n > 2 with                r ≤ d ≤ 2r, there exists a connected graph G with radG = r, diamG = d and st1x(G) = n for any vertex x in G. It is shown that for p,a and b are positive integers such that 4 ≤ a ≤ b ≤ p−1, then there exists a connected graph G of order p such that  (G) = a and  (G) = b for some vertex x in G.