THE EDGE FIXING EDGE-TO-VERTEX DETOUR NUMBER OF A GRAPH

We introduce the concept of the total edge fixing edge-to-vertex detour setof a connected graph . Let e be an edge of a graph . A set is called an edge fixing edge-to-vertex detour set of a connected graph  if every edge of lies on an detour, where . The edge fixing edge-to-vertex detour number  of  is the minimum cardinality of its edge fixing edge-to-vertex detour sets and any edge fixing edge-to-vertex detour set of cardinality    is an -set of Connected graphs of order p with edge fixing edge-to-vertex detour number 1 or   are characterized. Theedge fixing edge-to-vertex detour number for some standard graphs are determined.  It is shown that for every pair of positive integers with 2  there exists a connected graph  such that  and  for some edge .