Forbidden Induced Subgraphs for Bounded Shrub-Depth and the Expressive Power of MSO

TL;DR

This work provides a complete combinatorial–logical characterization of hereditary graph classes with bounded shrub-depth by explicit forbidden induced subgraphs, namely all flipped $H_t$ and flipped $tP_t$. It then ties these obstructions to model-theoretic stability notions, showing MSO-stability (and CMSO-stability, monadic variants) is equivalent to bounded shrub-depth, and that FO and MSO have identical expressive power precisely on these classes. A central mechanism is a suite of FO/MSO interpretations and transductions that embed paths into broader classes, enabling inexpressibility results and a nuanced separation of FO and MSO on unbounded shrub-depth classes. Collectively, the results close the gap on when FO and MSO coincide, resolving a question of Elberfeld–Grohe–Tantau and extending Gajarský–Hliněný by providing the missing obstruction-driven direction. The work further connects shrub-depth with stability theory through flip-flatness and path-interpretation techniques, and outlines avenues toward MSO-dependence as a parallel line of study with potential algorithmic implications.

Abstract

The graph parameter shrub-depth is a dense analog of tree-depth. We characterize classes of bounded shrub-depth by forbidden induced subgraphs. The obstructions are well-controlled flips of large half-graphs and of disjoint unions of many long paths. Applying this characterization, we show that on every hereditary class of unbounded shrub-depth, MSO is more expressive than FO. This confirms a conjecture of [Gajarský and Hliněný; LMCS 2015] who proved that on classes of bounded shrub-depth FO and MSO have the same expressive power. Combined, the two results fully characterize the hereditary classes on which FO and MSO coincide, answering an open question by [Elberfeld, Grohe, and Tantau; LICS 2012]. Our work is inspired by the notion of stability from model theory. A graph class C is MSO-stable, if no MSO-formula can define arbitrarily long linear orders in graphs from C. We show that a hereditary graph class is MSO-stable if and only if it has bounded shrub-depth. As a key ingredient, we prove that every hereditary class of unbounded shrub-depth FO-interprets the class of all paths. This improves upon a result of [Ossona de Mendez, Pilipczuk, and Siebertz; Eur. J. Comb. 2025] who showed the same statement for FO-transductions instead of FO-interpretations.

Figures (11)

• Figure 1: On the left: the half-graph of order $4$ (denoted as $H_4$) with vertices $\{a_1,\ldots,a_4,b_1,\ldots,b_4\}$ and edges between $a_i$ and $b_j$ if $i\leqslant j$. On the right: the $6$-vertex path (denoted as $P_6$).
• Figure 2: A flipped $3P_8$. More precisely the depicted graph is an $\mathcal{L}$-flip of $3P_8$ with $\mathcal{L} = \{L_1,\ldots,L_8\}$, where the following parts were flipped: $L_2$ with $L_3$ (red), $L_4$ with $L_7$ (blue), $L_7$ with $L_7$ (purple).
• Figure 3: All flipped $H_4$s (up to isomorphism). Figure replicated with permission from maehlmann-thesis.
• Figure 4: A map of \ref{['thm:main']}: combinatorial and logical characterizations of hereditary classes of bounded shrub-depth.
• Figure 5: From left to right: the star 4-crossing of order $3$, the clique 4-crossing of order $3$, and the rook graph of order $4$. Highlighted in blue we show how each of the three patterns of order $k$ contains an induced path on at least $k$ vertices.

Theorems & Definitions (102)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Definition 1.3
• Definition 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Lemma 2.1