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On the edge monophonic number of a graph
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For a connected graph G = (V, E), an edge monophonic set of G is a set M?
V(G) such that every edge of G is contained in a monophonic path joining some
pair of vertices in M. The edge monophonic number m1 (G) of G is the minimum
order of its edge monophonic sets and any edge monophonic set of order m1 (G)
is a minimum edge monophonic set of G. Connected graphs of order p with edge
monophonic number p are characterized. Necessary condition for edge
monophonic number to be p ? 1 is given. It is shown that for every two
integers a and b such that 2 ? a ? b, there exists a connected graph G with
m(G) = a and m1 (G) = b, where m(G) is the monophonic number of G.
Title: On the edge monophonic number of a graph
Description:
For a connected graph G = (V, E), an edge monophonic set of G is a set M?
V(G) such that every edge of G is contained in a monophonic path joining some
pair of vertices in M.
The edge monophonic number m1 (G) of G is the minimum
order of its edge monophonic sets and any edge monophonic set of order m1 (G)
is a minimum edge monophonic set of G.
Connected graphs of order p with edge
monophonic number p are characterized.
Necessary condition for edge
monophonic number to be p ? 1 is given.
It is shown that for every two
integers a and b such that 2 ? a ? b, there exists a connected graph G with
m(G) = a and m1 (G) = b, where m(G) is the monophonic number of G.
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