Finite and Algorithmic Model Theory

Algorithmic meta-theorems 247Example 5.7.3 Recall that a dominating set X inagraphG is a set X ⊆ V (G)such that for all v ∈ V (G), v ∈ X or there is a u ∈ X and {u, v} ∈E(G). Fork ∈ N, the formula( ∨ ( ) )ϕ k :=∃x 1 ...∃x k ∀y xi = y ∨ Eyx i1≤i≤kis true in a graph G if, and only if, G has a dominating set of size at most k.To convert this into an equivalent sentence in Gaifman normal form, wefirst observe that no connected graph of diameter at least 3k + 1 can have adominating set of size at most k. Here, the diameter of a graph is the maximumof the distance between any two vertices.Hence, on connected graphs, the formula ϕ k above is equivalent to theconjunction of the basic local sentenceψ := ¬∃x 1 ∃x 2 dist(x 1 ,x 2 ) > 3k + 1,saying that the diameter of G is greater than 3k + 1, and the basic localsentence ∃xχ(x), where χ(x) is the 3k + 1-local formula∃y 1 ∈ N 3k+1 (x) ...∃y k ∈ N 3k+1 (x)∀z ∈ N 3k+1 (x) ∨ ( )yi = z ∨ Ezy i .1≤i≤kNote that this formula correctly defines the existence of a dominating set ofsize k only in connected graphs, as in graphs with more than one componentthere may exist a dominating set of size k even though there are vertices x 1 ,x 2of distance greater than 3k + 1. Adapting the formula to this case requires alittle more effort.⊣5.7.2 First-Order Logic on Graphs of Bounded DegreeAs a first application of the use of Gaifman’s locality theorem for algorithmicmeta theorems we consider graph classes of bounded degree.Definition 5.7.4 AclassC of graphs has bounded degree if there is a d ∈ Nsuch that (G) ≤ d for all G ∈ C.In 1996, Seese [82] showed that model-checking for a fixed first-ordersentence can be done in linear time on graph classes of bounded degree.Theorem 5.7.5 (Seese [82]) For any class C of graphs of bounded degree andany fixed first-order sentence it can be decided in linear time whether G |= ϕfor a graph G ∈ C. In other words, first-order model-checking on C is fixedparametertractable by a linear fpt algorithm.