On γ-hyperellipticity of graphs

The basic objects of research in this paper are graphs and their branched coverings. By a graph, we mean a finite connected multigraph. The genus of a graph is defined as the rank of the first homology group. A graph is said to be gamma-hyperelliptic if it is a two fold branched covering of a genus gamma graph. The corresponding covering involution is called gamma-hyperelliptic. The aim of the paper is to provide a few criteria for the involution tau acting on a graph X of genus g to be gamma-hyperelliptic. If tau has at least one fixed point then the first criterium states that there is a basis in the homology group H-1 (X) whose elements are either invertible or split into gamma interchangeable pairs under the action of tau(*). The second criterium is given by the formula tr(H1(X)) (tau(*)) = 2 gamma - g. Similar results are also obtained in the case when tau acts fixed point free.