Let ???? be a connected graphand ???? ∈ ????(????). An edge fixing edge-to-vertex Steiner set ???? is called connected edge fixing edge-to-vertex Steiner set ???? if < ???? > is connected. The minimum cardinality of a connected edge fixing edge-to-vertex Steiner set of ???? is the connected edge fixing edge-to-vertex Steiner number of e of ???? and to denoted by ????????????????????????(????). The edge fixing edge-to-vertex Steiner set of ???? of ???? of cardinality ????????????????????????(????) is denoted by ????????????????????????-set of ????. Some general properties satisfied by this concept is studied. It is shown that for positive integers ????, ???? and ???? ≥ 2 with ???? ≤ ???? ≤ 2????, there exists a connected graph ???? with rad???? = ???? , diam???? = ???? and ????????????????????????(????) = ???? for some edge e in ????. For any positive integers ????and ???? with 2 ≤ ???? ≤ ????, there exists a connected graph ???? such that ????????????????(????) = a and ????????????????????????(????) = ???? for some edge e in ????.