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All Journal Electronic Journal of Graph Theory and Applications (EJGTA)
S. Athisayanathan
Department of Mathematics, St. Xavier's College (Autonomous), Palayamkottai-627 002, Tamil Nadu, India
Electrical & Electronics Engineering

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Weak edge triangle free detour number of a graph Sethu Ramalingam; S. Athisayanathan
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 10, No 2 (2022): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2022.10.2.22

Abstract

For any two vertices u and v in a connected graph G = (V, E), a u − v path P is called a u − v triangle free path if no three vertices of P induce a triangle. The triangle free detour distance D△f(u, v) is the length of a longest u − v triangle free path in G. A u − v path of length D△f(u, v) is called a u − v triangle free detour. A set S ⊆ V is called a weak edge triangle free detour set of G if every edge of G has both ends in S or it lies on a triangle free detour joining a pair of vertices of S. The weak edge triangle free detour number wdn△f(G) of G is the minimum order of its weak edge triangle free detour sets and any weak edge triangle free detour set of order wdn△f(G) is a weak edge triangle free detour basis of G. Certain properties of these concepts are studied. The weak edge triangle free detour numbers of certain classes of graphs are determined. Its relationship with the triangle free detour diameter is discussed and it is proved that for any three positive integers a, b and n of integers with 3 ≤ b ≤ n − a + 1 and a ≥ 4, there exists a connected graph G of order n with triangle free detour diameter D△f = a and wdn△f(G)=b. It is also proved that for any three positive integers a, b and c with 3 ≤ a ≤ b and c ≥ b + 2, there exists a connected graph G such that R△f = a, D△f = b and wdn△f(G)=c.
Co-Authors Sethu Ramalingam