On graphs determining links with maximal number of components via medial construction

Embedded graphs whose links have the largest possible number of components

“…In section 3 we study plane graphs. Many of our results replicate those in [3], although our emphasis is different because we are preparing to work on other surfaces.…”

“…The number µ is the same as the number of "straight-ahead" walks in medial graphs, as described in [8]. The focus in [3] is to characterize the plane graphs G whose µ(D(G)) is as large as possible, which is the cycle rank plus one; these are the "extremal" graphs. Maximising µ is also our principal interest here, although we will study graphs embedded on various orientable surfaces.…”

Extremal embedded graphs

“…In this section, we give some basic properties of µ(G), extending results in [7] and [9] from orientable ribbon graphs to non-orientable ones, and obtaining several new results at the same time. Otherwise, there are also two cases (see Figure 10).…”

“…A ribbon graph is said to be extremal if µ(G) = f (G) + γ(G) and a ribbon graph is plane if its Euler genus is zero. In [9], the authors studied extremal plane graphs and in [7], Huggett and Tawfik studied extremal cellularly embedded graphs on orientable surfaces of positive genus. In this paper we extend their results to any ribbon graph and also obtain that a ribbon graph is extremal if and only if its Petrial is plane.…”