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 ...

• Type: Journal
• Year: 2007
• Where: DAM
• Authors: Hajo Broersma, Fedor V. Fomin, Rastislav Kralovic, Gerhard J. Woeginger
• Keywords: DAM 2007, Hamiltonian Cycle, Parallel Knock-out Number, Parallel Knock-out Schemes