6 research outputs found
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On 3-Pushdown Graphs With Large Separators
For an integer s let ZS(n), the s-iterated logarithm function, be defined inductively: [O(n) = n, [8+1(n) = log2(l8(n)) for s 2:: o. We show that for every fixed s and all n large enough, there is an n-vertex 3-pushdown graph whose smallest separator contains at least n(n/[8(n)) vertices
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On Nontrivial Separators for k-Page Graphs and Simulations by Nondeterministic One-Tape Turing Machines
We show that the following statements are equivalent: 1. Statement 1. 3-pushdown graphs have sublinear separators. 2. Statement 1∗. k-page graphs have sublinear separators. 3. Statement 2. A one-tape nondeterministic Turing machine can simulate a two-tape machine in subquadratic time. None of the statements is known to be true or false at present. However, our proof of equivalence is quantitative-it relates exactly the separator size of the two kinds of graphs to the running time of the simulation in Statement 2. Using this equivalence we derive several graph-theoretic corollaries. There are known examples where upper bounds on graph properties imply upper bounds on computation time or space. There are other examples where lower bounds on graph properties are used to derive lower bounds on computation time in restricted settings. However, our results may constitute the first example where a graph problem is shown to be equivalent to a problem in computational complexity. In a companion paper we construct graphs and prove a lower bound or their separators. Using the equivalence we prove an almost linear lower bound for the size of separators for 3-pushdown graphs and an almost quadratic lower bound for simulating two-tape nondeterministic Turing machines by one-tape machines. Specifically, for an integers s let ls(n), the s-iterated logarithm function, be defined inductively: l°(n)=n, ls+1(n)=log2(ls(n)) for s⩾0. Then: 1. For every fixed s and all n, there is an n-vertex 3-pushdown graph whose smallest separator contains at least ω(n/ls(n)) vertices.2. There is a language L recognizable in real time by a two-tape nondeterministic Turing machine, but every on-line one-tape nondeterministic Turing machine that recognizes L requires ω(n2/ls(n)) time for any positive integer
Characterisations and Examples of Graph Classes with Bounded Expansion
Classes with bounded expansion, which generalise classes that exclude a
topological minor, have recently been introduced by Ne\v{s}et\v{r}il and Ossona
de Mendez. These classes are defined by the fact that the maximum average
degree of a shallow minor of a graph in the class is bounded by a function of
the depth of the shallow minor. Several linear-time algorithms are known for
bounded expansion classes (such as subgraph isomorphism testing), and they
allow restricted homomorphism dualities, amongst other desirable properties. In
this paper we establish two new characterisations of bounded expansion classes,
one in terms of so-called topological parameters, the other in terms of
controlling dense parts. The latter characterisation is then used to show that
the notion of bounded expansion is compatible with Erd\"os-R\'enyi model of
random graphs with constant average degree. In particular, we prove that for
every fixed , there exists a class with bounded expansion, such that a
random graph of order and edge probability asymptotically almost
surely belongs to the class. We then present several new examples of classes
with bounded expansion that do not exclude some topological minor, and appear
naturally in the context of graph drawing or graph colouring. In particular, we
prove that the following classes have bounded expansion: graphs that can be
drawn in the plane with a bounded number of crossings per edge, graphs with
bounded stack number, graphs with bounded queue number, and graphs with bounded
non-repetitive chromatic number. We also prove that graphs with `linear'
crossing number are contained in a topologically-closed class, while graphs
with bounded crossing number are contained in a minor-closed class
Unraveling k-page graphs
I show in this note that for any integer k and any k page graph G, there is an easily constructed 3-page graph G′ (called the unraveling of G) such that the minimum separator sizes of G and G′ are within a factor of k of each other. Further the maximum degree of a vertex of G′ is at most 2 plus the maximum degree of G