On the number of sources and terminals of sum-networks with capacity p/q

Rai et al. in [1] showed that any rational number p/q is the capacity of some sum-network. The authors proved this result by constructing such sum-networks. The authors first constructed a sum-network with capacity 1/q, and then they connected p of such networks in parallel to create a sum-network of capacity p/q. Such construction requires 2q² - q number of sources and 2q²-q number of terminals. In this paper, we ask the following question: is it possible to construct such sum-networks with lesser number of sources or terminals? The result in [2], where it is shown that there exists a k sources and 3 terminal sum-network with capacity k-1 over k; confirms that at least for some values of p and q, the answer is affirmative. For various values of p and q, we have made an attempt to construct sum-networks having lesser number of sources and terminals than used in [1], and have been able to construct various such sum-networks. In this paper, we present three such example sum-networks, with capacities 3 over 5; 4 over 7 and 5 over 12 respectively. However, we have not yet been able to come up with a systematic method to construct such sum-networks. We believe that studying these examples is a step towards constructing a sum-network, with capacity p over q, which requires minimum number of sources and terminals.