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We introduce a graph-parametric framework for obtaining obstruction characterizations of graph parameters with respect to partial ordering relations. For this, we define the notions of class obstruction, parametric obstruction, and universal obstruction as combinatorial objects that determine the asymptotic behavior of graph parameters. Our framework permits a unified framework for classifying graph parameters. Under this framework, we survey existing graph- theoretic results on most known graph parameters. Also we provide some unifying results on their classification.

We introduce the notion of a universal obstruction of a graph parameter with respect to some quasi-ordering relation on graphs. Universal obstructions may serve as a canonical obstruction characterization of the approximate behaviour of graph parameters. We provide an order-theoretic characterization of the finiteness of universal obstructions and, when this is the case, we present some algorithmic implications on the existence of fixed-parameter algorithms.

Many standard graph classes are known to be characterized by means of layouts (a permutation of its vertices) excluding some patterns. Important such graph classes are among others: proper interval graphs, interval graphs, chordal graphs, permutation graphs, (co-)comparability graphs. For example, a graph $G=(V,E)$ is a proper interval graph if and only if $G$ has a layout $L$ such that for every triple of vertices such that $x\prec_L y\prec_L z$, if $xz\in E$, then $xy\in E$ and $yz\in E$. Such a triple $x$, $y$, $z$ is called an indifference triple and layouts excluding indifference triples are known as indifference layouts. In this paper, we investigate the concept of tree-layouts. A tree-layout $T_G=(T,r,\rho_G)$ of a graph $G=(V,E)$ is a tree $T$ rooted at some node $r$ and equipped with a one-to-one mapping $\rho_G$ between $V$ and the nodes of $T$ such that for every edge $xy\in E$, either $x$ is an ancestor of $y$ or $y$ is an ancestor of $x$. Clearly, layouts are tree-layouts. Excluding a pattern in a tree-layout is defined similarly as excluding a pattern in a layout, but now using the ancestor relation. Unexplored graph classes can be defined by means of tree-layouts excluding some patterns. As a proof of concept, we show that excluding non-indifference triples in tree-layouts yields a natural notion of proper chordal graphs. We characterize proper chordal graphs and position them in the hierarchy of known subclasses of chordal graphs. We also provide a canonical representation of proper chordal graphs that encodes all the indifference tree-layouts rooted at some vertex. Based on this result, we first design a polynomial time recognition algorithm for proper chordal graphs. We then show that the problem of testing isomorphism between two proper chordal graphs is in P, whereas this problem is known to be GI-complete on chordal graphs., Comment: 33 pages, 13 figures

A strict bramble of a graph $G$ is a collection of pairwise-intersecting connected subgraphs of $G.$ The order of a strict bramble ${\cal B}$ is the minimum size of a set of vertices intersecting all sets of ${\cal B}.$ The strict bramble number of $G,$ denoted by ${\sf sbn}(G),$ is the maximum order of a strict bramble in $G.$ The strict bramble number of $G$ can be seen as a way to extend the notion of acyclicity, departing from the fact that (non-empty) acyclic graphs are exactly the graphs where every strict bramble has order one. We initiate the study of this graph parameter by providing three alternative definitions, each revealing different structural characteristics. The first is a min-max theorem asserting that ${\sf sbn}(G)$ is equal to the minimum $k$ for which $G$ is a minor of the lexicographic product of a tree and a clique on $k$ vertices (also known as the lexicographic tree product number). The second characterization is in terms of a new variant of a tree decomposition called lenient tree decomposition. We prove that ${\sf sbn}(G)$ is equal to the minimum $k$ for which there exists a lenient tree decomposition of $G$ of width at most $k.$ The third characterization is in terms of extremal graphs. For this, we define, for each $k,$ the concept of a $k$-domino-tree and we prove that every edge-maximal graph of strict bramble number at most $k$ is a $k$-domino-tree. We also identify three graphs that constitute the minor-obstruction set of the class of graphs with strict bramble number at most two. We complete our results by proving that, given some $G$ and $k,$ deciding whether ${\sf sbn}(G) \leq k$ is an ${\sf NP}$-complete problem.