Hypermaps: Partial duality and polynomial invariant

Hypermaps are combinatorial structures that extend maps by allowing hyperedges to connect more than two vertices. Chmutov and Vignes-Tourneret introduced the concept of partial duality in hypermaps, defining it in three ways: involutions on the flag set, permutations on the flag set, and edge [Formula: see text]-colored graphs. In our research, we redefine partial duality related to the hyperedge set using three permutations on a finite set, Chmutov's arrow presentation method, and the bipartite graph associated with a hypermap. We then introduce the partial duality polynomial for hypermaps and look at some of its properties.