166 research outputs found
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
A linear-time algorithm for finding a complete graph minor in a dense graph
Let g(t) be the minimum number such that every graph G with average degree
d(G) \geq g(t) contains a K_{t}-minor. Such a function is known to exist, as
originally shown by Mader. Kostochka and Thomason independently proved that
g(t) \in \Theta(t*sqrt{log t}). This article shows that for all fixed \epsilon
> 0 and fixed sufficiently large t \geq t(\epsilon), if d(G) \geq
(2+\epsilon)g(t) then we can find this K_{t}-minor in linear time. This
improves a previous result by Reed and Wood who gave a linear-time algorithm
when d(G) \geq 2^{t-2}.Comment: 6 pages, 0 figures; Clarification added in several places, no change
to arguments or result
- …