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Connected Edge Fixed Monophonic Number of a Graph |
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PP: 2357-2364 |
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doi:10.18576/amis/100639
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Author(s) |
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P. Titus,
S. Eldin Vanaja,
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Abstract |
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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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