A structure theorem for rooted connectivity in bidirected graphs

Abstract: Recently, bidirected graphs have received increasing attention from the graph theory community with both structural and algorithmic results. Bidirected graphs are a generalization of directed graphs, consisting of an undirected graph together with a map assigning each endpoint of every edge either sign $+$ or $-$. The connectivity properties of bidirected graphs are more complex than those of directed graphs and not yet well understood. In this paper, we show a structure theorem about rooted connectivity in bidirected graphs in terms of directed graphs. As applications, we prove Lov\'asz' flame theorem, Pym's theorem and a strong variant of Menger's theorem for a class of bidirected graphs and provide counterexamples in the general case.