The Vertex Detour Monophonic Number of a Graph

Abstract

In this paper we determine bounds for x-detour monophonic number and characterize graphs which realize these bounds. A connected graph of order p with vertex detour monophonic numbers either p-1 or p-2 for every vertex is characterized. It is shown
that for each triple a; b and p of integers with 1 ≤ a ≤ b ≤ p - 4, there is a connected graph G of order p such that x-monophonic number is a and x-detour monophonic number is b for some vertex x in G. Also, for integers a, b and p with 1 ≤ a ≤ p - b and b ≥ 2, there is a connected graph G of order p such that x-detour monophonic number is a and monophonic eccentricity of x is b for some vertex x in G.

Key words and phrases. monophonic path, detour monophonic path, vertex monophonic number, vertex detour monophonic number.

2000 Mathematics Subject Classification. 05C12