Dominating cycles in halin graphs

A cycle in a graph is dominating if every vertex lies at distance at most one from the cycle and a cycle is D -cycle if every edge is incident with a vertex of the cycle. In this paper, first we provide recursive formulae for finding a shortest dominating cycle in a Halin graph; minor modifications can give formulae for finding a shortest D -cycle. Then, dominating cycles and D -cycles in a Halin graph H are characterized in terms of the cycle graph, the intersection graph of the faces of H .