The Descriptive Complexity of Subgraph Isomorphism Without Numerics

Abstract

Let F be a connected graph with ℓ vertices. The existence of a subgraph isomorphic to F can be defined in first-order logic with quantifier depth no better than ℓ, simply because no first-order formula of smaller quantifier depth can distinguish between the complete graphs K ℓ and K ℓ−1 . We show that, for some F , the existence of an F subgraph in sufficiently large connected graphs is definable with quantifier depth ℓ − 3. On the other hand, this is never possible with quantifier depth better than ℓ/2. If we, however, consider definitions over connected graphs with sufficiently large treewidth, the quantifier depth can for some F be arbitrarily small comparing to ℓ but never smaller than the treewidth of F . Moreover, the definitions over highly connected graphs require quantifier depth strictly more than the density of F . Finally, we determine the exact values of these descriptive complexity parameters for all connected pattern graphs F on 4 vertices.

Tight Bounds on the Asymptotic Descriptive Complexity of Subgraph Isomorphism

“…We begin with a few simple properties of graphs without S q,p subgraphs. The following lemma generalizes a property of S 4,4 -free graphs observed in [17]. Proof.…”

“…In our earlier paper [17], we found an example of a pattern graph F with D ′ (F ) ≤ ℓ − 3 and observed, on the other hand, that W ′ (F ) ≥ 1 2 ℓ − 1 2 for all F . It remained unknown whether the difference between W ′ (F ) and W (F ) = ℓ could be arbitrarily large.…”

“…From (15)- (17) we conclude that F contains a pendant star F 1 , a pendant path F 2 , and a pendant sparkler F 3 such that each of these subgraphs has more than 1 3 ℓ + 1 vertices. If F 1 and F 3 are vertex-disjoint or share only one vertex, which must be the central vertex of F 1 and the end vertex of the tail part of F 3 , these subgraphs together occupy more than 2 3 ℓ + 1 vertices, and then there remains not enough space for the pendant path F 2 .…”

“…Simplifying the general notation introduced in Subsection 2.1, we write D ′ (F ) = D ′ (S(F )) and W ′ (F ) = W ′ (S(F )). We now state simple combinatorial bounds for these parameters that were used already in [17].…”

“…As we already mentioned, D ′ (P 3 ) = W ′ (P 3 ) = 1 by trivial reasons. The equalities D ′ (P 4 ) = 3 and W ′ (P 4 ) = 2 are proved in [17]. We, therefore, suppose that ℓ ≥ 5.…”