The upper open monophonic number of a graph

A. P. Santhakumaran

M. Mahendran

Hindustan University

India

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ABSTRACT

For a connected graph G of order n,a subset S of vertices of G is a monophonic set of G if each vertex v in G lies on a x-y monophonic path for some elements x and y in S. The minimum cardinality of a monophonic set of G is defined as the monophonic number of G, denoted by m(G). A monophonic set of cardinality m(G) is called a m-set of G.A set S of vertices of a connected graph G is an open monophonic set of G if for each vertex v in G ,either v is an extreme vertex of G and v G S,or v is an internal vertex of a x-y mono-phonic path for some x,y G S. An open monophonic set of minimum cardinality is a minimum open monophonic set and this cardinality is the open monophonic number, om(G). An open monophonic set S of vertices in a connected graph G is a minimal open monophonic .set if no proper subset of S is an open monophonic set of G.The upper open monophonic number om+ (G) is the maximum cardinality of a minimal open monophonic set of G. The upper open monophonic numbers of certain standard graphs are determined. It is proved that for a graph G of order n, om(G) = n if and only if om+(G)= n. Graphs G with om(G) = 2 are characterized. If a graph G has a minimal open monophonic set S of cardinality 3, then S is also a minimum open monophonic set of G and om(G) = 3. For any two positive integers a and b with 4 < a < b, there exists a connected graph G with om(G) = a and om+(G) = b.

2010 Mathematics Subject Classification. 05C12,05C70.

Key Words: Distance, geodesic, geodetic number, open geodetic number, monophonic number, open monophonic number, upper open monophonic number.

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A. P. Santhakumaran

Department of Mathematics Hindustan University

Hindustan Institute of Technology and Science Padur

Chennai-603 103,

India

e-mail : apskumar1953@yahoo.co.in

M. Mahendran

Department of Mathematics Hindustan University

Hindustan Institute of Technology and Science Padur

Chennai-603 103,

India

e-mail : magimani83@gmail.com

Received : February 2014. Accepted : October 2014