For a connected graph G=(V,E)G=(V,E) of order at least two, a chord of a path PP is an edge joining two non-adjacent vertices of PP. A path PP is called a monophonic path if it is a chordless path. A monophonic set of GG is a set SS of vertices such that every vertex of GG lies on a monophonic path joining some pair of vertices in SS. The monophonic number of GG is the minimum cardinality of its monophonic sets and is denoted by m(G)m(G). A geodetic set of GG is a set SS of vertices such that every vertex of GG lies on a geodesic joining some pair of vertices in SS. The geodetic number of GG is the minimum cardinality of its geodetic sets and is denoted by g(G)g(G). The number of extreme vertices in GG is its extreme order ex(G)ex(G). A graph GG is an extreme monophonic graph if m(G)=ex(G)m(G)=ex(G) and an extreme geodesic graph if g(G)=ex(G)g(G)=ex(G). Extreme monophonic graphs of order pp with monophonic number pp and p−1p−1 are characterized. It is shown that every pair a,ba,b of integers with 0≤a≤b0≤a≤b is realized as the extreme order and monophonic number, respectively, of some graph. For positive integers r,dr,d and k≥3k≥3 with r<dr<d, it is shown that there exists an extreme monophonic graph GG of monophonic radius rr, monophonic diameter dd, and monophonic number kk. Also, we give a characterization result for a graph GG which is both extreme geodesic and extreme monophonic.
