A graph with dangling edges can be simply a bipartitioned graph where the set of vertices splits into a union $\ V:=A\cup B\ $ of disjoint sets $\ A\ B, $ and vertices $\ b\in B\ $ can have at the most two neighbors (from $\ A).\ $

The partition into $\ A\ B\ $ has to be specified in the general case if we want a unique representation of the bi-partitioned graph as a graph with dangling edges so that vertices from $\ B\ $ would be interpreted as the dangling edges.