Two-hop Realizability

The Two-hop problem (THP) (also known as the minimum forwarding set problem) is to select a minimum set of one-hop neighbors of a given source node to broadcast messages to all of its two-hop neighbors. Despite the simplicity of THP, its computation complexity is unknown. Consider a bipartite graph B(A,B, E) such that the degree of any node in A is 3 and degree of any node in B is 2. If B can be embedded in the plane such that it represents an instance of a THP, then we can conclude the computational complexity of THP. In this paper we show that a bipartite graph, in which the degree of any node in B is less than or equal to two can be embedded in the plane such that it represents an instance of a THP.