Excluded Grid Theorem Research Articles

On the tree-width of even-hole-free graphs

The class of all even-hole-free graphs has unbounded tree-width, as it contains all complete graphs. Recently, a class of (even-hole, K4)-free graphs was constructed, that still has unbounded tree-width (Sintiari and Trotignon, 2019). The class has unbounded degree and contains arbitrarily large clique-minors. We ask whether this is necessary.We prove that for every graph G, if G excludes a fixed graph H as a minor, then G either has small tree-width, or G contains a large wall or the line graph of a large wall as induced subgraph. This can be seen as a strengthening of Robertson and Seymour’s excluded grid theorem for the case of minor-free graphs. Our theorem implies that every class of even-hole-free graphs excluding a fixed graph as a minor has bounded tree-width. In fact, our theorem applies to a more general class: (theta, prism)-free graphs. This implies the known result that planar even hole-free graphs have bounded tree-width (Silva et al., 2010).We conjecture that even-hole-free graphs of bounded degree have bounded tree-width. If true, this would mean that even-hole-freeness is testable in the bounded-degree graph model of property testing. We prove the conjecture for subcubic graphs and we give a bound on the tree-width of the class of (even hole, pyramid)-free graphs of degree at most 4.

On Tseitin Formulas, Read-Once Branching Programs and Treewidth

We show that any nondeterministic read-once branching program that decides a satisfiable Tseitin formula based on an n × n grid graph has size at least 2Ω(n). Then using the Excluded Grid Theorem by Robertson and Seymour we show that for an arbitrary graph G(V, E) any nondeterministic read-once branching program that computes a satisfiable Tseitin formula based on G has size at least $2^{\Omega (\text {tw}(G)^{\delta })}$ for all δ < 1/36, where tw(G) is the treewidth of G (for planar graphs and some other classes of graphs the statement holds for δ = 1). We apply the mentioned results to the analysis of the complexity of derivations in the proof system OBDD(∧,reordering) and show that any OBDD(∧,reordering)-refutation of an unsatisfiable Tseitin formula based on a graph G has size at least $2^{\Omega (\text {tw}(G)^{\delta })}$ . We also show an upper bound O(|E|2pw(G)) on the size of OBDD representations of a satisfiable Tseitin formula based on G and an upper bound $O(|E||V| 2^{\text {pw}(G)}+|\text {TS}_{G,c}|^{2})$ on the size of OBDD(∧)-refutation of an unsatisifable Tseitin formula TSG, c, where pw(G) is the pathwidth of G.

Towards tight(er) bounds for the Excluded Grid Theorem

We study the Excluded Grid Theorem, a fundamental structural result in graph theory, that was proved by Robertson and Seymour in their seminal work on graph minors. The theorem states that there is a function f:Z+→Z+, such that for every integer g>0, every graph of treewidth at least f(g) contains the (g×g)-grid as a minor. For every integer g>0, let f(g) be the smallest value for which the theorem holds. Establishing tight bounds on f(g) is an important graph-theoretic question. Robertson and Seymour showed that f(g)=Ω(g2log⁡g) must hold. For a long time, the best known upper bounds on f(g) were super-exponential in g. The first polynomial upper bound of f(g)=O(g98polylog⁡g) was proved by Chekuri and Chuzhoy. It was later improved to f(g)=O(g36polylog⁡g), and then to f(g)=O(g19polylog⁡g). In this paper we further improve this bound to f(g)=O(g9polylog⁡g). We believe that our proof is significantly simpler than the proofs of the previous bounds. Moreover, while there are natural barriers that seem to prevent previous methods from yielding tight bounds for the theorem, it seems conceivable that the techniques proposed in this paper can lead to even tighter bounds on f(g).

Highly Connected Sets and the Excluded Grid Theorem

We present a short proof of the excluded grid theorem of Robertson and Seymour, the fact that a graph has no large grid minor if and only if it has small tree-width. We further propose a very simple obstruction to small tree-width inspired by that proof, showing that a graph has small tree-width if and only if it contains no large highly connected set of vertices.