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