### The forcing monophonic and forcing geodetic numbers of a graph

#### Abstract

For a connected graph *G* = (*V*, *E*), let a set *S* be a *m*-set of *G*. A subset *T* ⊆ *S* is called a forcing subset for *S* if *S* is the unique *m*-set containing *T*. A forcing subset for S of minimum cardinality is a minimum forcing subset of *S*. The forcing monophonic number of S, denoted by *fm*(*S*), is the cardinality of a minimum forcing subset of *S*. The forcing monophonic number of *G*, denoted by fm(G), is *fm*(*G*) = min{*fm*(*S*)}, where the minimum is taken over all minimum monophonic sets in G. We know that *m*(*G*) ≤ *g*(*G*), where *m*(*G*) and *g*(*G*) are monophonic number and geodetic number of a connected graph *G* respectively. However there is no relationship between *fm*(*G*) and *fg*(*G*), where *fg*(*G*) is the forcing geodetic number of a connected graph *G*. We give a series of realization results for various possibilities of these four parameters.

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PDFDOI: http://dx.doi.org/10.19184/ijc.2020.4.2.5

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