Characterization of Upper Detour Monophonic Domination Number

Abstract

This paper introduces the concept of upper detour monophonic domination number of a graph. For a connected graph G with vertex set V ( G ) , a set M ⊆ V ( G ) is called minimal detour monophonic dominating set, if no proper subset of M is a detour monophonic dominating set. The maximum cardinality among all minimal monophonic dominating sets is called upper detour monophonic domination number and is denoted by γ + dm ( G ) . For any two positive integers p and q with 2 ≤ p ≤ q there is a connected graph G with γ m ( G ) = γ dm ( G ) = p and γ + dm ( G ) = q . For any three positive integers p, q, r with 2 < p < q < r , there is a connected graph G with m ( G ) = p , γ dm ( G ) = q and γ + dm ( G ) = r . Let p and q be two positive integers with 2 < p < q such that γ dm ( G ) = p and γ + dm ( G ) = q . Then there is a minimal DMD set whose cardinality lies between p and q . Let p, q and r be any three positive integers with 2 ≤ p ≤ q ≤ r . Then, there exist a connected graph G such that γ dm ( G ) = p, γ + dm ( G ) = q and | V ( G ) | = r . γ + dm ( G ) = q . Entonces existe un conjunto DMD mínimo cuya cardinalidad se encuentra entre p y q . Sean p, q y r tres enteros positivos cualquiera con 2 ≤ p ≤ q ≤ r . Entonces existe un grafo conexo G tal que γ dm ( G ) = p, γ + dm ( G ) = q y | V ( G ) | = r .