The Complexity of Growing a Graph

We introduce here the model of growing graphs, a model of dynamic networks in which nodes can generate new nodes, thus expanding the network. This motivates the algorithmic problem of constructing a target graph G, starting from a single node. To properly model this, we assume that every node u can generate at most one node v in every round (or time slot). Every newly generated node v can activate edges with other nodes, only at the time of its birth, provided that these nodes are up to a small distance d away from v. We show that the most interesting case is when d=2. As we prove, in order to achieve the construction of a target graph G in a small number of time slots, we might need to pay for auxiliary edges (the "excess edges"), which will be eventually removed. This creates a trade-off between the number of time slots and the number of excess edges required to construct a target graph. In this paper, we deal with the following algorithmic question: Given a target graph G of n nodes, can G be constructed in at most k time slots and with at most \ell excess edges? On the positive side, we provide polynomial-time algorithms that efficiently construct fundamental graph families, such as lines, stars, trees, and planar graphs. In particular, we show that trees can be constructed in a poly-logarithmic number of slots with linearly many excess edges, while planar graphs can be constructed in a logarithmic number of slots with O(n\log n) excess edges. We also give a polynomial-time algorithm for deciding whether a graph can be constructed in \log n slots with \ell = 0 excess edges. On the negative side, we prove that the problem of determining the minimum number of slots required for a graph to be constructed with zero excess edges (i) is NP-complete and (ii) for any \varepsilon>0, cannot be approximated within n^{1-\varepsilon}, unless P=NP.