The total edge Steiner number of a graph

Abstract

A total edge Steiner set of G is an edge Steiner set W such that the subgraph <W> induced by has no isolated vertex. The minimum cardinality of a total edge Steiner set of G is the total edge Steiner number of G and is denoted by ste(G). Some general properties satisfied by this concept are studied. The total edge Steiner numbers of certain classes of graphs is studied. Connected graphs of order p with total edge Steiner number 2 or 3 are characterized. Necessary conditions for total edge Steiner number to be p or p − 1 is given. It is shown that for every pair a and b of integers with 2 ≤ a < b and b > a+ 1, there exists a connected graph G such that se(G) = a and ste(G) = b. Also it shown that for every pair a and b of integers with 4 ≤ a < b and b > a + 1, there exists a connected graph G such that st(G) = a and ste(G) = b.