In this paper, we study the complexity of the chip-firing reachability problem. We show that for Eulerian digraphs, the reachability problem can be decided in strongly polynomial time, even if the digraph has multiple edges. We also show a special case when the reachability problem can be decided in polynomial time for general digraphs: if the target distribution is recurrent restricted to each strongly connected component. As a further positive result, we show that the chip-firing reachability problem is in co-NP for general digraphs. We also show that the chip-firing halting problem is in co-NP for Eulerian digraphs.
Given a set of planar curves (Jordan arcs), each pair of which meets -either crosses or touches -exactly once, we establish an upper bound on the number of touchings.We show that such a curve family has O (t 2 n) touchings, where t is the number of faces in the curve arrangement that contains at least one endpoint of one of the curves. Our method relies on finding special subsets of curves called quasi-grids in curve families; this gives some structural insight into curve families with a high number of touchings.
Baker and Norine proved a Riemann-Roch theorem for divisors on undirected graphs. The notions of graph divisor theory are in duality with the notions of the chip-firing game of Björner, Lovász and Shor. We use this connection to prove Riemann-Roch-type results on directed graphs. We give a simple proof for a Riemann-Roch inequality on Eulerian directed graphs, improving a result of Amini and Manjunath. We also study possibilities and impossibilities of Riemann-Roch-type equalities in strongly connected digraphs and give examples. We intend to make the connections of this theory to graph theoretic notions more explicit via using the chip-firing framework.
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