A polynomial excluded-minor approximation of treedepth

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×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 Ω(kc) has at least one of the following minors (each of treedepth at least k): • a k×k grid, • a complete binary tree of height k, or • a path of order 2k. 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(kc). 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.

Duke Scholars

Author Benjamin Rossman Computer Science