The Total Detour Number of a Graph

Abstract

For a connected graph G = (V, E), aset S ⊆ V(G) is called a total detour set of G if S is a detour set of G and the subgraph G[S] induced by S has no isolated vertex. The total detour number tdn(G) of G is the minimum order of its total detour sets and any total detour set of order tdn(G) is called a td-set of G. The total detour number of some standard graphs are determined. Necessary conditions for the total detour number to be 2 or P are given. It is shown that for any three integers a, b and p with 2 ≤ a < b ≤ p, there exists a connected graph G of order p with tdn (G) = a and cdn (G) = b. Also it is shown that for each triple D, k, p of integers with 3 ≤ k ≤ p - D + 1 and D ≥ 4, there exists a connected graph G of order P with detour diameter D and tdn (G) = k. A total detour set S in a connected graph G is called a minimal total detour set of G if no proper subset of S is a total detour set of G. The upper total detour number tdn+ (G) of G is the maximum cardinality of a minimal total detour set of G. It is shown that for every pair a, b of integers with 3 ≤ a < b, there exists a connected graph G with tdn (G) = a and tdn+ (G) = b.