Let k be an algebraically closed field of characteristic $$p>0$$ p>0 . Let and be two smooth proper connected curves, each endowed with an automorphism $$\sigma _i:B_i \rightarrow B_i$$ σi:Bi→Bi of order p. Let , and let $$\sigma :Y \rightarrow Y$$ σ:Y→Y be the automorphism . We show that the graph of the resolution of any singularity of is a star-shaped graph with three terminal chains when $$B_2$$ B2 is an ordinary curve of positive genus. The intersection matrix N of the resolution satisfies , and can be completely determined when $$B_1$$ B1 is also ordinary, or when $$\sigma _1$$ σ1 has a unique fixed point. The singularity is rational. Wild -quotient singularities of surfaces are expected to have resolution graphs which are trees, with associated intersection matrices N satisfying for some $$r\geqslant 0$$ r⩾0 . We show, for any $$s>0$$ s>0 coprime to p, the existence of resolution graphs with one node, $$s+2$$ s+2 terminal chains, and with intersection matrix N satisfying .