tt-sails and sparse hereditary classes of unbounded tree-width

It has long been known that the following basic objects are obstructions to bounded tree-width: for arbitrarily large $t$, $(1)$ the complete graph $K_t$, $(2)$ the complete bipartite graph $K_{t,t}$, $(3)$ a subdivision of the $(t \times t)$-wall and $(4)$ the line graph of a subdivision of the $(t \times t)$-wall. We now add a further \emph{boundary object} to this list, a subdivision of a \emph{$t$-sail}. These results have been obtained by studying sparse hereditary \emph{path-star} graph classes, each of which consists of the finite induced subgraphs of a single infinite graph whose edges can be decomposed into a path (or forest of paths) with a forest of stars, characterised by an infinite word over a possibly infinite alphabet. We show that a path-star class whose infinite graph has an unbounded number of stars, each of which connects an unbounded number of times to the path, has unbounded tree-width. In addition, we show that such a class is not a subclass of circle graphs, a hereditary class whose unavoidable induced subgraphs with large treewidth were identified by Hickingbotham, Illingworth, Mohar and Wood \cite{hickingbotham:treewidth_circlegraphs:}. We identify a collection of \emph{nested} words with a recursive structure that exhibit interesting characteristics when used to define a path-star graph class. These graph classes do not contain any of the four basic obstructions but instead contain graphs that have large tree-width if and only if they contain arbitrarily large subdivisions of a $t$-sail. Furthermore, like classes of bounded degree or classes excluding a fixed minor, these sparse graph classes do not contain a minimal class of unbounded tree-width

Infinitely many minimal classes of graphs of unbounded clique-width

© 2017 Elsevier B.V. The celebrated theorem of Robertson and Seymour states that in the family of minor-closed graph classes, there is a unique minimal class of graphs of unbounded tree-width, namely, the class of planar graphs. In the case of tree-width, the restriction to minor-closed classes is justified by the fact that the tree-width of a graph is never smaller than the tree-width of any of its minors. This, however, is not the case with respect to clique-width, as the clique-width of a graph can be (much) smaller than the clique-width of its minor. On the other hand, the clique-width of a graph is never smaller than the clique-width of any of its induced subgraphs, which allows us to be restricted to hereditary classes (that is, classes closed under taking induced subgraphs), when we study clique-width. Up to date, only finitely many minimal hereditary classes of graphs of unbounded clique-width have been discovered in the literature. In the present paper, we prove that the family of such classes is infinite. Moreover, we show that the same is true with respect to linear clique-width

MSO Undecidability for some Hereditary Classes of Unbounded Clique-Width

Seese's conjecture for finite graphs states that monadic second-order logic (MSO) is undecidable on all graph classes of unbounded clique-width. We show that to establish this it would suffice to show that grids of unbounded size can be interpreted in two families of graph classes: minimal hereditary classes of unbounded clique-width; and antichains of unbounded clique-width under the induced subgraph relation. We explore a number of known examples of the former category and establish that grids of unbounded size can indeed be interpreted in them.Comment: 23 page

Stable graphs of bounded twin-width

We prove that every class of graphs $\mathscr C$ that is monadically stable and has bounded twin-width can be transduced from some class with bounded sparse twin-width. This generalizes analogous results for classes of bounded linear cliquewidth and of bounded cliquewidth. It also implies that monadically stable classes of bounded twin-widthare linearly $\chi$-bounded.Comment: 44 pages, 2 figure

Graph parameters and the speed of hereditary properties

In this thesis we study the speed of hereditary properties of graphs and how this defines some of the structure of the properties. We start by characterizing several graph parameters by means of minimal hereditary classes. We then give a global characterization of properties of low speed, before looking at properties with higher speeds starting at the Bell number. We then introduce a new parameter, clique-width, and show that there are an infinite amount of minimal hereditary properties with unbounded clique-width. We then look at the factorial layer in more detail and focus on P7-free bipartite graphs. Finally we discuss word-representable graphs