Deciding First-Order Properties for Sparse Graphs

Abstract

Abstract-We present a linear-time algorithm for deciding first-order logic (FOL) properties in classes of graphs with bounded expansion. Many natural classes of graphs have bounded expansion: graphs of bounded tree-width, all proper minor-closed classes of graphs, graphs of bounded degree, graphs with no subgraph isomorphic to a subdivision of a fixed graph, and graphs that can be drawn in a fixed surface in such a way that each edge crosses at most a constant number of other edges. We also develop an almost linear-time algorithm for deciding FOL properties in classes of graphs with locally bounded expansion; those include classes of graphs with locally bounded tree-width or locally excluding a minor.More generally, we design a dynamic data structure for graphs belonging to a fixed class of graphs of bounded expansion. After a linear-time initialization the data structure allows us to test an FOL property in constant time, and the data structure can be updated in constant time after addition/deletion of an edge, provided the list of possible edges to be added is known in advance and their addition results in a graph in the class. In addition, we design a dynamic data structure for testing existential properties or the existence of short paths between prescribed vertices in such classes of graphs. All our results also hold for relational structures and are based on the seminal result of Nešetřil and Ossona de Mendez on the existence of low tree-depth colorings.

Practical algorithms for MSO model-checking on tree-decomposable graphs

“…Dawar, Grohe, and Kreutzer gave polynomial time algorithms for graphs locally excluding an arbitrary but fixed minor [53]. These results were generalized by Dvořák, Král, and Thomas [59] to graph classes of bounded expansion and locally bounded expansion. In particular, all graph classes of bounded treewidth, all proper minor-closed graph classes, and all graph classes of bounded degree have bounded expansion; all graph classes of bounded local treewidth and those locally excluding a minor have locally bounded expansion [59].…”

“…These results were generalized by Dvořák, Král, and Thomas [59] to graph classes of bounded expansion and locally bounded expansion. In particular, all graph classes of bounded treewidth, all proper minor-closed graph classes, and all graph classes of bounded degree have bounded expansion; all graph classes of bounded local treewidth and those locally excluding a minor have locally bounded expansion [59]. All these graph classes are furthermore closed under taking subgraphs and nowhere dense, a notion for sparse graph classes introduced by Nešetřil and Ossona de Mendez [142,143].…”

“…The latter result can probably not be extended any further: it is known that under standard complexity theoretic assumptions a corresponding result for classes of graphs that are closed under taking subgraphs and are somewhere dense is unlikely [94,59,119].…”

“…Theorem 10 (Dvořák, Král, and Thomas [59]). Fix an FO-definable graph property Π and a graph class C of bounded expansion.…”

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].…”