Efficient reversal of transductions of sparse graph classes

TL;DR

The paper addresses reversing first-order transductions on sparse graph classes by introducing an efficient sparsification that maps any monadically stable class with inherently linear neighborhood complexity to a bounded-expansion class via FO-interpretable recovery. The core idea is to construct a sparse graph H containing the original vertex set plus a rooted tree T of bounded height, such that adjacency in the original graph G is recoverable by a fixed FO-interpretation I from H; the sparsified class D = {S(G)} is shown to have bounded expansion and is transducible from the original class C. A central technical contribution is a main lemma that, given a bipartite graph G in C, computes a partition of one side into parts aligned with leaders on the other side, with bounded overlap and FO-definability, enabling the T- and H- constructions. Together with a novel direct approach to neighborhood covers, these ingredients yield an O(n^4) sparsification algorithm and establish an equivalence between structurally bounded expansion and the combination of monadic stability with linear neighborhood complexity. The results have algorithmic implications for FO model-checking and related FO-query tasks, since preprocessing to a bounded-expansion class allows leveraging known FPT or linear-time FO techniques, and the output being transducible from the input preserves structural properties across the transformation.

Abstract

(First-order) transductions are a basic notion capturing graph modifications that can be described in first-order logic. In this work, we propose an efficient algorithmic method to approximately reverse the application of a transduction, assuming the source graph is sparse. Precisely, for any graph class $\mathcal{C}$ that has structurally bounded expansion (i.e., can be transduced from a class of bounded expansion), we give an $O(n^4)$-time algorithm that given a graph $G\in \mathcal{C}$, computes a vertex-colored graph $H$ such that $G$ can be recovered from $H$ using a first-order interpretation and $H$ belongs to a graph class $\mathcal{D}$ of bounded expansion. This answers an open problem raised by Gajarský et al. In fact, for our procedure to work we only need to assume that $\mathcal{C}$ is monadically stable (i.e., does not transduce the class of all half-graphs) and has inherently linear neighborhood complexity (i.e., the neighborhood complexity is linear in all graph classes transducible from $\mathcal{C}$). This renders the conclusion that the graph classes satisfying these two properties coincide with classes of structurally bounded expansion.

Figures (4)

• Figure 1: For given vertices $u,v \in V(G)$, we display a node $p$ adjacent to both in $H$, such that no child of $p$ is adjacent to both. There are an even number of vertices on the path from the root to $p$, and thus $u,v$ are adjacent in $G$.
• Figure 2: Visualization of the output of \ref{['lem:main']}. We can choose $k^* = 3$ and verify that, for example, the red vertex is adjacent to at most 3 parts from $\mathcal{F}$, also marked red.
• Figure 3: Visualization of step $i$ of the algorithm. As depicted, each set $P_j$ fully adjacent to $a_j$, and fully non-adjacent to $a_{j+1},a_{j+2}$, etc. Vertex $a_i$ and its highlighted partner are $1$-near-twins, disagreeing only on the first set (from the left) of $B_{i-1}$. As softly depicted at the bottom, $\mathcal{B}_i$ is obtained from $\mathcal{B}_{i-1}$ by removing the first set (as $a_i$ was its only neighbor), and merging the second and third set (as $a_i$ was its only distinguishing vertex).
• Figure 4: A left branching of a bipartite graph into three branches $G_1,G_2,G_3$ with respective leaders $\ell_1,\ell_2,\ell_3$. As required, each edge is in at least one branch, and each right vertex is in exactly one branch. Each left vertex is in at most two branches, and thus the branching has overlap two.

Theorems & Definitions (69)

• Theorem 1
• Corollary 2
• Lemma 2
• Definition 3
• Definition 4
• Definition 5
• Remark 6
• Definition 7
• Definition 8
• Lemma 9