The k disjoint shortest paths problem (k-DSPP) on a graph with k source-sink pairs (s i , t i ) asks for the existence of k pairwise edge-or vertex-disjoint shortest s i -t i -paths. It is known to be NP-complete if k is part of the input. Restricting to 2-DSPP with strictly positive lengths, it becomes solvable in polynomial time. We extend this result by allowing zero edge lengths and give a polynomial time algorithm based on dynamic programming for 2-DSPP on undirected graphs with non-negative edge lengths.

An r-gentiling is a dissection of a shape into r ≥ 2 parts which are all similar to the original shape. An r-reptiling is an r-gentiling of which all parts are mutually congruent. By applying gentilings recursively, together with a rule that defines an order on the parts, one may obtain an order in which to traverse all points within the original shape. We say such a traversal is a facecontinuous space-filling curve if, at any level of recursion, the interior of the union of any set of consecutive parts is connected-that is, consecutive parts must always meet along an edge. Most famously, the isosceles right triangle admits a 2-reptiling, which forms the basis of the face-continuous Sierpiński space-filling curve; many other right triangles admit reptilings and gentilings that yield face-continuous space-filling curves as well. In this study we investigate what acute triangles admit non-trivial reptilings and gentilings, and whether these can form the basis for face-continuous spacefilling curves. We derive several properties of reptilings and gentilings of acute (sometimes also obtuse) triangles, leading to the following conclusion: no face-continuous space-filling curve can be constructed on the basis of reptilings of acute triangles.

In this paper we focus on r-neighbor bootstrap percolation, which is a process on a graph where initially a set A0 of vertices gets infected. Now subsequently, an uninfected vertex becomes infected if it is adjacent to at least r infected vertices. Call A f the set of vertices that is infected after the process stops. More formally set At :At. We deal with finite graphs only and denote by n the number of vertices. We are mainly interested in the size of the final set A f . We present a theorem for degenerate graphs that bounds the size of the final infected set. More precisely for a d-degenerate graph, if r > d, we bound the size set A f from above by (1 + d r−d )|A0|.arXiv:1605.07002v1 [math.CO]

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