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.

DOI : http://dx.doi.org/10.29055/jcms/1004

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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