The Complexity of Growing a Graph

We study a new algorithmic process of graph growth. The process starts from a single initial vertex u 0 and operates in discrete time-steps, called slots . In every slot t ≥ 1, the process updates the current graph instance to generate the next graph instance G t , according to the following vertex and edge update rules. The process first sets G t = G t − 1 . Then, for every u ∈ V ( G t − 1 ), it adds at most one new vertex u (cid:48) to V ( G t ) and adds the edge uu (cid:48) to E ( G t ) alongside any subset of the edges { vu (cid:48) | v ∈ V ( G t − 1 ) is at distance at most d − 1 from u in G t − 1 } , for some integer d ≥ 1 fixed in advance. The process completes slot t after removing any (possibly empty) subset of edges from E ( G t ). Removed edges are called excess edges . G t is the graph grown by the process after t slots. The goal of this paper is to investigate the algorithmic and structural properties of this process of graph growth. , process. We show that the most interesting is when d = 2 and that there is a natural trade-off between k and (cid:96) . We by investigating growth schedules of (cid:96) = 0 excess edges. On the positive side, we provide polynomial-time algorithms that decide whether a graph has growth schedules of k = log n or k = n − 1 slots. Along the way, interesting connections to cop-win graphs are being revealed. On the negative side, we establish strong hardness results for the problem of determining the minimum number of slots required to grow a graph with zero excess edges. In particular, we show that the problem (i) is NP-complete and (ii) for any ε > 0, cannot be approximated within n 1 − ε , unless P = NP. We then move our focus to the other extreme of the ( k, (cid:96) )-spectrum, to investigate growth schedules of (poly)logarithmic slots. We show that trees can be grown in a polylogarithmic number of slots using linearly many excess edges, while planar graphs can be grown in a logarithmic number of slots using O ( n log n ) excess edges. We also give lower bounds on the number of excess edges, when the number of slots is fixed to log n .