### 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

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DOI: http://dx.doi.org/10.23755/rm.v44i0.903

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