Applied Mathematics & Information Sciences

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For an edge xy in a connected graph G of order p ≥ 3, a set S ⊆V(G) is an xy-monophonic set of G if each vertex v ∈V(G) lies on either an x−u monophonic path or a y−u monophonic path for some element u in S. The minimum cardinality of an xymonophonic set of G is defined as the xy-monophonic number of G, denoted by mxy(G). An xy-monophonic set of cardinality mxy(G) is called a mxy-set of G. A connected xy-monophonic set of G is an xy-monophonic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected xy-monophonic set of G is the connected xy-monophonic number of G and is denoted by cmxy(G). A connected xy-monophonic set of cardinality cmxy(G) is called a cmxy-set of G. We determine bounds for it and find the same for some special classes of graphs. If d, n and p ≥ 4 are positive integers such that 2 ≤ d ≤ p−2 and 1 ≤ n ≤ p−1, then there exists a connected graph G of order p, monophonic diameter d and cmxy(G) = n for some edge xy in G. Also, we give some characterization and realization results for the parameter cmxy(G).

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