Characterization of Upper Detour Monophonic Domination Number

• M. Mohammed Abdul Khayyoom Department of Mathematics PTM Govt. College, Perintalmanna, Malappuram, Kerala, India.
Keywords: Monophonic number, Domination Number, Detour monophonic number, Detour monophonic domination number, 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 $$M\subseteq V(G)$$  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 $$\gamma_{dm}^+(G)$$. For any two positive integers $$p$$ and $$q$$  with $$2 \leq p \leq q$$ there is a connected graph $$G$$ with $$\gamma_m (G) = \gamma_{dm}(G) = p$$ and $$\gamma_{dm}^+(G)=q$$. For any three positive integers $$p, q, r$$ with $$2 < p < q < r$$, there is a connected graph $$G$$ with $$m(G) = p$$, $$\gamma_{dm}(G) = q$$ and $$\gamma_{dm}^+(G)= r$$. Let $$p$$ and $$q$$ be two positive integers with $$2 < p<q$$ such that $$\gamma_{dm}(G) = p$$ and $$\gamma_{dm}^+(G)= q$$. 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 $$2 \leq p \leq q \leq r$$. Then, there exist a connected graph $$G$$ such that $$\gamma_{dm}(G) = p , \gamma_{dm}^+(G)= q$$ and $$\lvert V(G) \rvert = r$$.

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Published
2020-12-07
How to Cite
[1]
M. M. Abdul Khayyoom, “Characterization of Upper Detour Monophonic Domination Number”, CUBO, A Mathematical Journal, vol. 22, no. 3, pp. 315–324, Dec. 2020.
Section
Articles