1Assistant Professor,
Department of Mathematics,
Rani Anna Govt. College (W), Tirunelveli – 627 008,
Affiliated to Manonmaniam Sundaranar University, Tirunelveli, INDIA.
2 Research Scholar,
Department of Mathematics,
Rani Anna Govt. College (W), Tirunelveli – 627 008,
Affiliated to Manonmaniam Sundaranar University, Tirunelveli, INDIA.

ABSTRACT

In this paper the concept of total edge detour monophonic number M of a
graph G is introduced. For a connected graph G = (V, E) of order at least two, a total
edge detour monophonic set M of a graph G is an edge detour monophonic set such
that either M = V or the sub graph induced by M has no isolated vertices. The
minimum cardinality of a total edge detour monophonic set of G is the total edge
detour monophonic number of G and is denoted by edmt(G). We determine bounds
for it and characterize graphs which realize these bounds. It is shown that if every
positive integers p, a and b such that 3 ≤ a ≤ b ≤ p − 2, then there exists a
connected graph G of order p, edm(G) = a and edmt(G) = b. If p and k are
positive integers such that 3 ≤ k ≤ p, then there exists a connected graph G of order
p, edmt(G) = k. For positive integers a,b such that 4 ≤ a ≤ b with b ≤ 2a there
exists a connected graph G such that edm(G) = a and edmt(G) = b.
AMS Subject classification : 05C12.

Keywords :Detour monophonic set, detour monophonic number, edge detour
monophonic set and edge detour monophonic number, total edge detour monophonic
set and total edge detour monophonic number.