43889321
It is tough to be a plumber
In the Linux computer game KPlumber, the objective is to rotate tiles in a raster of squares so as to complete a system of pipes. We give a complexity classification for the original game and various special cases of it that arise from restricting the set of six possible tiles.Most of the cases are NP-complete. One polynomially solvable case is settled by formulating it as a perfect matching problem; other polynomial cases are settled by simple sweepline techniques. Moreover, we show that all the unsettled cases are polynomial time equivalent.
22610152
Counting k-Subsets and Convex k-gons in the Plane
We consider problems of the following type: Let S be a set of n points in the plane, let p be some point in S and let K be an integer, 3≤k≤n. We show how to compute the overall number of convex k-gons with corners at points in S, for any fixed k≥4.
15492927
Counting Convex Polygons in Planar Point Sets
Given a set S of n points in the plane, we compute in time O(n3) the total number of convex polygons whose vertices are a subset of S. We give an O(m · n3) algorithm for computing the number of convex k-gons with vertices in S, for all values k = 3,…, m; previously known bounds were exponential (O(nk/2). We also compute the number of empty convex polygons (resp.,k-gons, k m) with vertices in S in time O(n3) (resp., O(m · n3)).
29745112
A note on the bottleneck graph partition problem
10937579
Eliminating graphs by means of parallel knock-out schemes
In 1997 Lampert and Slater introduced parallel knock-out schemes, an iterative process on graphs that goes through several rounds. In each round of this process, every vertex eliminates exactly one of its neighbors. The parallel knock-out number of a graph is the minimum number of rounds after which all vertices have been eliminated (if possible). The parallel knock-out number is related to well-known concepts like perfect matchings, hamiltonian cycles, and 2-factors. We derive a number of combinatorial and algorithmic results on parallel knock-out numbers: for families of sparse graphs (like planar graphs or graphs of bounded tree-width), the parallel knock-out number grows at most logarithmically with the number n of vertices; this bound is basically tight for trees. Furthermore, there is a family of bipartite graphs for which the parallel knock-out number grows proportionally to the square root of n. We characterize trees with parallel knock-out number at most 2, and we show that the parallel knock-out number for trees can be computed in polynomial time via a dynamic programming approach (whereas in general graphs this problem is known to be NP-hard). Finally, we prove that the parallel knock-out number of a claw-free graph is either infinite or less than or equal to 2.
39921005
Graph colorings
From the combination of knowledge and actions, someone can improve their skill and ability. It will lead them to live and work much better. This is why, the students, workers, or even employers should have reading habit for books. Any book will give certain knowledge to take all benefits. This is what this graph colorings tells you. It will add more knowledge of you to life and work better. Try it and prove it.