eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-08-16
40:1
40:16
10.4230/LIPIcs.CSL.2017.40
article
On the First-Order Complexity of Induced Subgraph Isomorphism
Verbitsky, Oleg
Zhukovskii, Maksim
Given a graph F, let I(F) be the class of graphs containing F
as an induced subgraph. Let W[F] denote the minimum k such that
I(F) is definable in k-variable first-order logic. The recognition
problem of I(F), known as Induced Subgraph Isomorphism (for the pattern graph F), is solvable in time O(n^{W[F]}). Motivated by this fact, we are interested in determining or estimating the value of W[F]. Using Olariu's characterization of paw-free graphs, we show that I(K_3+e) is definable by a first-order sentence of quantifier depth 3, where K_3+e denotes the paw graph. This provides an example of a graph F with W[F] strictly less than the number of vertices in F.
On the other hand, we prove that W[F]=4 for all F on 4 vertices
except the paw graph and its complement. If F is a graph on t vertices, we prove a general lower bound W[F]>(1/2-o(1))t, where the function in the little-o notation approaches 0 as t increases. This bound holds true even for a related parameter W^*[F], which is
defined as the minimum k such that I(F) is definable in the k-variable
infinitary logic. We show that W^*[F] can be strictly less than W[F]. Specifically, W^*[P_4]=3 for P_4 being the path graph on 4 vertices.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol082-csl2017/LIPIcs.CSL.2017.40/LIPIcs.CSL.2017.40.pdf
the induced subgraph isomorphism problem
descriptive and computational complexity
finite-variable first-order logic
quantifier depth and variable w