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- W3186151750 abstract "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 $$tge 1$$ , the process updates the current graph instance to generate the next graph instance $$G_t$$ . The process first sets $$G_t = G_{t-1}$$ . Then, for every $$uin V(G_{t-1})$$ , it adds at most one new vertex $$u^prime $$ to $$V(G_{t})$$ and adds the edge $$uu^prime $$ to $$E(G_{t})$$ alongside any subset of the edges $${vu^prime ;|; vin V(G_{t-1})$$ is at distance at most $$d-1$$ from u in $$G_{t-1}}$$ , for some integer $$dge 1$$ 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. Graph Growth Problem: Given a graph family F, we are asked to design a centralized algorithm that on any input target graph $$Gin F$$ , will output such a process growing G, called a growth schedule for G. Additionally, the algorithm should try to minimize the total number of slots k and of excess edges $$ell $$ used by the process. We show that the most interesting case is when $$d = 2$$ and that there is a natural trade-off between k and $$ell $$ . We begin by investigating growth schedules of $$ell =0$$ 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 $$varepsilon >0$$ , cannot be approximated within $$n^{frac{1}{3}-varepsilon }$$ , unless P = NP. We then move our focus to the other extreme of the $$(k,ell )$$ -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 $$O(nlog n)$$ excess edges. We also give lower bounds on the number of excess edges, when the number of slots is fixed to $$log n$$ ." @default.
- W3186151750 created "2021-08-02" @default.
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- W3186151750 date "2022-01-01" @default.
- W3186151750 modified "2023-09-24" @default.
- W3186151750 title "The Complexity of Growing a Graph" @default.
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- W3186151750 doi "https://doi.org/10.1007/978-3-031-22050-0_9" @default.
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