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For any vertex x in a connected graph G of order p ≥ 2, a set S ⊆ V (G) is an x-monophonic set of G if each vertex v ∈ V (G) lies on an x − y monophonic path for some element y in S. The minimum cardinality of an x-monophonic set of G is defined as the x-monophonic number of G, denoted by mx(G). An x-monophonic set S is called a minimal x-monophonic set if no proper subset of S is an x-monophonic set. The upper x-monophonic number, denoted by m + x (G), is defined as the maximum cardinality ofdoi:10.12732/ijpam.v106i2.4 fatcat:4ajfdu7i2vfhzczkxdnmh7hgsm