# IJPAM: Volume 106, No. 2 (2016)

**THE UPPER VERTEX MONOPHONIC NUMBER OF A GRAPH**

Department of Mathematics

University College of Engineering Nagercoil

Anna University

Tirunelveli Region, Nagercoil, 629 004, INDIA

Department of Mathematics

V.V. College of Engineering

Tirunelveli, 627 657, INDIA

**Abstract. **For any vertex in a connected graph of order , a set
is an x-monophonic set of if each vertex lies on an monophonic path for some element in . The minimum cardinality of an -monophonic set of is defined as the x-monophonic number of , denoted by . An -monophonic set is called a minimal x-monophonic set if no proper subset of is an -monophonic set. The upper x-monophonic number, denoted by , is defined as the maximum cardinality of a minimal -monophonic set of . We determine bounds for it and find the same for some special classes of graphs. For any two positive integers and with
, there exists a connected graph with and for some vertex in . Also, it is shown that for any three positive integers , and with and , there exists a connected graph with , and a minimal -monophonic set of cardinality .

**Received: **October 9, 2015

**AMS Subject Classification: **05C12

**Key Words and Phrases: **monophonic path, vertex monophonic set, vertex monophonic number, minimal vertex monophonic set, upper vertex monophonic number

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**DOI: 10.12732/ijpam.v106i2.4**

International Journal of Pure and Applied Mathematics

**How to cite this paper?****Source:****ISSN printed version:**1311-8080

**ISSN on-line version:**1314-3395

**Year:**2016

**Volume:**106

**Issue:**2

**Pages:**389 - 400

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**This work is licensed under the Creative Commons Attribution International License (CC BY).**