We introduce a new notion of graph sparsification based on spectral similarity of graph Laplacians: spectral sparsification requires that the Laplacian quadratic form of the sparsifier approximate that of the original. This is equivalent to saying that the Laplacian of the sparsifier is a good preconditioner for the Laplacian of the original. We prove that every graph has a spectral sparsifier ...

A linear k-forest of an undirected graph G is a subgraph of G whose components are paths with lengths at most k. The linear k-arboricity Of G, denoted by lak(G), is the minimum number of linear k-forests needed to partition the edge set E(G) of G. In case that the lengths of paths are not restricted, we then have the linear arboricity ofG, denoted by la(G). In this paper, the exact values of th...

Lexicographic depth first search (LexDFS) is a graph search protocol which has already proved to be a powerful tool on cocomparability graphs, a subfamily of perfect graphs. Cocomparability graphs have been well studied through their complements (comparability graphs) and their corresponding posets. Recently however LexDFS has led to a number of elegant near linear time algorithms on cocomparab...

We show that if G is a Ramsey size-linear graph and x,y 2 V(G ) then if we add a sufficiently long path between x and y we obtain a new Ramsey size-linear graph. As a consequence we show that if G is any graph such that every cycle in G contains at least four consecutive vertices of degree 2 then G is Ramsey size-linear. ß 2002 John Wiley & Sons, Inc. J Graph Theory 39: 1–5, 2002