2008/03/09 by Fedor V. Fomin, Fomin, Fedor V., Yngve Villanger +1 · 3 citations
Computer Science · Mathematics · #Advanced Graph Theory Research #Graph Labeling and Dimension Problems #Limits and Structures in Graph Theory #cs.DS
paper · pdf · doi:10.48550/arxiv.0803.1321
Corrected typos
arxiv created 2008/05/05 · arxiv updated 2009/12/01
For a given graph G and integers b,f >= 0, let S be a subset of vertices of G of size b+1 such that the subgraph of G induced by S is connected and S can be separated from other vertices of G by removing f vertices. We prove that every graph on n vertices contains at most n\binomb+fb such vertex subsets. This result from extremal combinatorics appears to be very useful in the design of several enumeration and exact algorithms. In particular, we use it to provide algorithms that for a given n-vertex graph G - compute the treewidth of G in time O(1.7549n) by making use of exponential space and in time O(2.6151n) and polynomial space; - decide in time O(((2n+k+1)/(3))k+1⋅ kn6) if the treewidth of G is at most k; - list all minimal separators of G in time O(1.6181n) and all potential maximal cliques of G in time O(1.7549n). This significantly improves previous algorithms for these problems.