Drawing combinatorial structures is a new challenge aimed to a better control of data in industry and in business as well. Speaking of graph we often understand map since a rotation system of edges around the vertices is specified by the context or by choice of the data structures. Search is the crucial step in the process of drawing. Depth-First Search has generated in the past well known algorithms for planarity testing, map transformation, graph drawing. Breadth-First Search on the counterpart was rather neglected and appears to be the right tool for coding and drawing maps. A bijection is proposed here, between the pointed maps with m edges and the connected involutions on {0,1,...,2m+1}.