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We develop a general diagrammatic theory of welded graphs, and provide anextension of Satoh's Tube map from welded graphs to ribbon surface-links. As atopological application, we obtain a complete link-homotopy classification ofso-called \emph{knotted punctured spheres} in $4$--space, by means of the$4$--dimensional Milnor invariants introduced by the authors in \cite{cutAMY}.On the algebraic side, we show that the theory of welded graphs can bereinterpreted as a theory of Wirtinger group presentations, up to a natural setof transformations; these groups arise as the fundamental group of the exteriorof the surface-link obtained from the given welded graph by the extended Tubemap. Finally, we address the injectivity question for the Tube map, identifyinga new family of local moves on welded links, called $\Upsilon$ moves, underwhich the (non extended) Tube map is invariant.

Welded graphs, Wirtinger groups and knotted punctured spheres