Tree-width and large grid minors in planar graphs
نویسنده
چکیده
We call H a minor of a graph G if H is obtainable from a subgraph of G by edge contractions. If g ≥ 2, the g-grid is the simple graph with vertices vij (1 ≤ i, j ≤ g) where vij and vi′j′ are adjacent if |i− i′|+ |j − j′| = 1; see Robertson and Seymour (1991). A tree-decomposition of a graph G , is a pair (T, S), where T is a tree and S = {St : t ∈ V (T )} is a family of subsets of V (G), called bags, such that
منابع مشابه
An Estimate of the Tree-Width of a Planar Graph Which Has Not a Given Planar Grid as a Minor
We give a more simple than in [8] proof of the fact that if a finite graph has no minors isomorphic to the planar grid of the size of r × r, then the tree-width of this graph is less than exp(poly(r)). In the case of planar graphs we prove a linear upper bound which improves the quadratic estimate from [5].
متن کاملTree-width and the monadic quantifier hierarchy
It is well known that on classes of graphs of bounded tree-width, every monadic second-order property is decidable in polynomial time. The converse is not true without further assumptions. It follows from the work of Robertson and Seymour, that if a class of graphs K has unbounded tree-width and is closed under minors, then K contains all planar graphs. But on planar graphs, three-colorability ...
متن کاملOperations which preserve path-width at most two
The number of excluded minors for the graphs with path-width at most two is too large. To give a practical characterization of the obstructions for path-width at most two, we introduce the concept reducibility. We describe some operations, which preserve path-width at most two, and reduce the excluded minors to smaller graphs. In this sense, there are ten graphs which are non-reducible and obst...
متن کاملGraph minor hierarchies
In their work on graph minors, Robertson and Seymour begin by describing graphs whose structure is particularly simple, graphs that look roughly like thickened paths [ 8, 3 ]. They say that such graphs have small ‘path-width’. Of course, not every graph looks roughly like a thickened path, but it is possible to describe those that do not: every graph of large path-width contains a particular su...
متن کاملClique Minors in Cartesian Products of Graphs
A clique minor in a graph G can be thought of as a set of connected subgraphs in G that are pairwise disjoint and pairwise adjacent. The Hadwiger number η(G) is the maximum cardinality of a clique minor in G. It is one of the principle measures of the structural complexity of a graph. This paper studies clique minors in the Cartesian product G H. Our main result is a rough structural characteri...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Discrete Mathematics & Theoretical Computer Science
دوره 13 شماره
صفحات -
تاریخ انتشار 2011