The Upper and Forcing Fault Tolerant Geodetic Number of a Graph

Abstract

A fault tolerant geodetic  is said to be minimal fault tolerant geodetic set of  if no proper subset of  is a fault tolerant geodetic set of  is called the upper fault tolerant geodetic number of  is denoted by . Some general properties satisfied by this concept are studied. For connected graphs of order  with to be  is given. It is shown that for every pair of with , there exists a connected graph such that  and , where  is the fault tolerant geodetic number of  and  is the upper fault tolerant geodetic number of a graph. Let S be a -set of . A subset  is called a forcing subset for  if  is the unique -set containing T. A forcing subset for  of minimum cardinality is a minimum forcing subset of . The forcing fault tolerant geodetic number of S, denoted by, is the cardinality of a minimum forcing subset of . The forcing fault tolerant geodetic number of , denoted by  is , where the minimum is taken over all -sets in . The forcing fault tolerant geodetic number of some standard graphs are determined. Some of its general properties are studied. It is shown that for every pair of positive integers  and  with  and  there exists a connected graph  such that  and