1,979 research outputs found
Grad and Classes with Bounded Expansion II. Algorithmic Aspects
Classes of graphs with bounded expansion are a generalization of both proper
minor closed classes and degree bounded classes. Such classes are based on a
new invariant, the greatest reduced average density (grad) of G with rank r,
∇r(G). These classes are also characterized by the existence of several
partition results such as the existence of low tree-width and low tree-depth
colorings. These results lead to several new linear time algorithms, such as an
algorithm for counting all the isomorphs of a fixed graph in an input graph or
an algorithm for checking whether there exists a subset of vertices of a priori
bounded size such that the subgraph induced by this subset satisfies some
arbirtrary but fixed first order sentence. We also show that for fixed p,
computing the distances between two vertices up to distance p may be performed
in constant time per query after a linear time preprocessing. We also show,
extending several earlier results, that a class of graphs has sublinear
separators if it has sub-exponential expansion. This result result is best
possible in general
Grad and classes with bounded expansion I. decompositions
We introduce classes of graphs with bounded expansion as a generalization of
both proper minor closed classes and degree bounded classes. Such classes are
based on a new invariant, the greatest reduced average density (grad) of G with
rank r, grad r(G). For these classes we prove the existence of several
partition results such as the existence of low tree-width and low tree-depth
colorings. This generalizes and simplifies several earlier results (obtained
for minor closed classes)
Largest reduced neighborhood clique cover number revisited
Let be a graph and . The largest reduced neighborhood clique
cover number of , denoted by , is the largest, overall
-shallow minors of , of the smallest number of cliques that can cover
any closed neighborhood of a vertex in . It is known that
, where is an incomparability graph and is
the number of leaves in a largest shallow minor which is isomorphic to an
induced star on leaves. In this paper we give an overview of the
properties of including the connections to the greatest
reduced average density of , or , introduce the class
of graphs with bounded neighborhood clique cover number, and derive a simple
lower and an upper bound for this important graph parameter. We announce two
conjectures, one for the value of , and another for a
separator theorem (with respect to a certain measure) for an interesting class
of graphs, namely the class of incomparability graphs which we suspect to have
a polynomial bounded neighborhood clique cover number, when the size of a
largest induced star is bounded.Comment: The results in this paper were presented in 48th Southeastern
Conference in Combinatorics, Graph Theory and Computing, Florida Atlantic
University, Boca Raton, March 201
- …