On monophonic position sets in graphs

General Position Polynomials

“…Several variations of the general position number have been investigated in the literature. For example, a set S ⊆ V (G) is in monophonic position if no induced path of G contains three vertices of S (see [24]), whilst S is a mutualvisibility set if for any u, v ∈ S there exists a shortest u, v-path in G that does not pass through S \ {u, v} (see [9]). We suggest than an interesting direction for future research would be to explore the polynomials counting such sets and their relation to the general position polynomials.…”

Lower general position sets in graphs

“…The edge version of the general position problem has also been recently studied in [30]. A related problem is the monophonic position problem obtained by replacing "shortest path" in the general position problem by "induced path", see [39]. Another variant of the general position problem is the mutual-visibility problem that asks for a largest set of vertices S, such that for each pair of vertices in S there is a shortest path connecting them that does not contain a third vertex of S, see [15].…”

“…In this section we relate the lower general position number to the monophonic position number mentioned in Section 1. The monophonic position number was introduced in [39] as follows. A path P in a graph G is induced or monophonic if G contains no chords between non-consecutive vertices of P .…”

“…A set M ⊆ V (G) is in monophonic position if no induced path in G contains three or more vertices of M ; the monophonic position number mp(G) of G is the number of vertices in a largest monophonic position set. It was shown in [39] that for any graph G we have mp(G) ≤ gp(G) and that for any 2 ≤ a ≤ b there exists a graph with mp(G) = a and gp(G) = b. This question was explored further in [41], which asked for the smallest possible order of a graph G with mp(G) = a and gp(G) = b for given a ≤ b.…”

On some extremal position problems for graphs

“…In [7,8] various problems in geodetic convexity in graphs were generalised to monophonic paths. In the paper [17] the present authors introduced the 'monophonic position number' of a graph, which is defined similarly to the gp-number, but with 'shortest path' replaced by 'induced path'; a set S of vertices in a graph G is in monophonic position if there is no monophonic path in G that contains more than two elements of S. A set satisfying this condition is called a monophonic position set or simply an mp-set. The mp-and gp-numbers of trees have a particularly simple form.…”