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