# IJPAM: Volume 80, No. 3 (2012)

**THE UPPER HULL NUMBER OF A GRAPH**

Department of Mathematics

Government College of Engineering

Tirunelveli, 627 007, INDIA

Department of Mathematics

Cape Institute of Technology

Levengipuram, 627114, INDIA

**Abstract. **For a connected graph , the hull number of a graph is the minimum cardinality of a set of vertices whose convex hull contains all vertices of . A hull set in a connected graph is called a minimal hull set of if no proper subset of is a hull set of . The upper hull number of is the maximum cardinality of a minimal hull set of . Connected graphs of order with upper hull number or are characterized. It is shown that for every integer , there exists a connected graph with and . A graph is an extreme hull graph if , that is if has a unique minimum hull set consisting of the extreme vertices of . It is shown that for every pair , of integers with
, there exists a connected extreme hull graph such that
and , where is the geodetic number of a graph.

**Received: **March 11, 2012

**AMS Subject Classification: **05C12

**Key Words and Phrases: **hull number, upper hull number, geodetic number

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**Source:**International Journal of Pure and Applied Mathematics

**ISSN printed version:**1311-8080

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

**Year:**2012

**Volume:**80

**Issue:**3