A polynomial excluded-minor approximation of treedepth

Abstract

Treedepth is a minor-monotone graph invariant in the family of “width measures” that includes treewidth and pathwidth. The characterization and approximation of these invariants in terms of excluded minors has been a topic of interest in the study of sparse graphs. A celebrated result of Chekuri and Chuzhoy (2014) shows that treewidth is polynomially approximated by the largest $k\times k$ grid minor in a graph. In this paper, we give an analogous polynomial approximation of treedepth via three distinct obstructions: grids, balanced binary trees, and paths. Namely, we show that there is a constant $c$ such that every graph with treedepth $\Omega (k^c)$ has at least one of the following minors (each of treedepth at least $k$):

• a $k\times k$ grid,
• a complete binary tree of height $k$, or
• a path of order $2^k$.

Moreover, given a graph $G$ we can, in randomized polynomial time, find an embedding of one of these minors or conclude that treedepth of $G$ is at most $O(k^c)$. This result has applications in various settings where bounded treedepth plays a role. In particular, we describe one application in finite model theory, an improved homomorphism preservation theorem over finite structures [Rossman, 2017], which was the original motivation for our investigation of treedepth.