Abstract

Total Edge Detour Monophonic Number of a Graph

P. Arul Paul Sudhahar1 and M. Little Flower*2

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.

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