Locally Excluding a Minor

Abstract

We introduce the concept of locally excluded minors. Graph

Structural Properties of Sparse Graphs

“…This is a strengthening of the result proved in [9]. Clearly, quasi-wide quasi-wide classes of graphs include classes of graphs locally excluding a minor [29]. Using the theory developed for classes of sparse graphs we shall give a complete characterization of hereditary classes of graphs which are wide, almost wide and quasi-wide.…”

“…They were studied extensively, see e.g. [29]. A "standard example" of a class with bounded local expansion is the class G of graphs G such that girth(G) ≥ ∆(G): consider any fixed integer r and the subgraph G v of G ∈ G induced by the r-neighborhood of v. Either ∆(G) < 2r and thus |G v | ≤ (2r) r or ∆(G) > 2r thus girth(G) > 2r hence G v is a tree.…”

“…Both these questions were intensively studied in the unrestricted cases [97,105,7] as well as under various restrictions (to minor closed classes and classes of bounded degree graphs; see e.g. [8,10,27,29,35,58,84]. This research continued by considering classes of bounded expansion and, more recently, classes of nowhere dense graphs.…”

Graph Minors and Parameterized Algorithm Design

“…The part of G that is inside P is the compass K of W which can be seen as the union of a collection of graphs (flaps) that are tree-like (have bounded treewidth) and are "planted" in that territory. Theorem 12 was used also in [2,24] with the name "the Trinity Lemma". However, a more depictive alternative nomenclature might be the "Sunny Forest Lemma", in the sense that the compass K is a forest, whose trees are the flaps, and X is the sun throwing its rays at it!…”

Locality of Queries Definable in Invariant First-Order Logic with Arbitrary Built-in Predicates

“…For instance, any non-local property such as acyclicity, connectivity, or k-colorability can immediately be shown non-expressible in a logic that exposes a certain amount of locality (see, e.g., [12,Chapter 4]). Locality is also exploited in an essential way in the design of efficient algorithms for evaluating firstorder definable queries on certain classes of structures [3,5].…”