Recognizing weighted directed cartesian graph bundles

In this paper we show that methods for recognizing Cartesian graph bundles can be generalized to weighted digraphs. The main result is an algorithm which lists the sets of degenerate arcs for all representations of digraph as a weighted directed Cartesian graph bundle over simple base digraphs not containing transitive tournament on three vertices. Two main notions are used. The first one is the new relation ~δ∗ defined among the arcs of a digraph as a weighted directed analogue of the well-known relation δ∗. The second one is the concept of half-convex subgraphs. A subgraph H is half-convex in G if any vertex x ∈ G \H has at most one predecessor and at most one successor.