Decomposing graphs into stable and ordered parts

TL;DR

This work addresses how dependent hereditary graph classes can be generated from simpler bases via first-order transductions by introducing modelizations that couple posets with colored graphs. The authors establish a strong result for the first non-trivial case of bounded linear cliquewidth, showing that such classes admit modelizations in a monadically dependent coupling of disjoint unions of chains and colored graphs with bounded pathwidth, and extend this to bounded-size bounded linear cliquewidth decompositions with colored graphs of bounded expansion. Key constructions include semi-plane rooted trees, sobs, cotrees/bicotrees, and amalgams that enable decompositions and modelizations to be carried through gluing and harvesting of bounded parameters. The findings provide a normal-form perspective that connects stability/dependence with order-like and tree-like structures, suggesting broad conjectures about universal decompositions for dependent graphs and implications for FO model checking and transductions. The results advance a program linking monadic notions of stability and dependence to graph parameters such as pathwidth and treewidth, with potential impact on algorithmic graph theory and the theory of transductions.

Abstract

Connections between structural graph theory and finite model theory recently gained a lot of attention. In this setting, many interesting questions remain on the properties of dependent (NIP) hereditary classes of graphs, in particular related to first-order transductions. In this paper, we study modelizations (which are strong forms of transduction pairings) of classes of graphs by classes of structures. In particular, we consider models obtained by coupling a partial order and a colored graph (thus forming a partially ordered colored graph). Motivated by Simon's decomposition theorem of dependent types into a stable part and a distal (order-like) part, we conjecture that every dependent hereditary class of graphs admits a modelization in a monadically dependent coupling of a class of posets with bounded treewidth cover graphs and a monadically stable class of colored graphs. In this paper, we consider the first non-trivial case (classes with bounded linear cliquewidth) and prove that the conjecture holds in a strong form, the model class being a monadically dependent coupling of a class of disjoint unions of chains and a class of colored graphs with bounded pathwidth. We extend our study to classes that admit bounded-size bounded linear cliquewidth decompositions and prove that they have a modelization in a monadically dependent coupling of a class of disjoint unions of chains and a class of colored graphs with bounded expansion, the model class also admitting bounded-size bounded linear cliquewidth decompositions.

Figures (5)

• Figure 1: A semi-plane rooted tree. The larger node is the root $r$. The parent function is denoted by $\pi$. The ancestor relation (which is a tree-order) is denoted by $\prec$. For example, $v\prec w$. Green (triangular) vertices are chain vertices (marked $C$), red (square) vertices are antichain vertices (marked $A$), and gray (circle) vertices are leaves (marked $L$). The children of a chain vertex are ordered by $\mathrel{\vartriangleleft}$. For example, the children of $v$ are ordered by $x_1\mathrel{\vartriangleleft} x_2\mathrel{\vartriangleleft} x_3\mathrel{\vartriangleleft} x_4\mathrel{\vartriangleleft} x_5$ and the children of the root are ordered by $\pi(u)\mathrel{\vartriangleleft} v\mathrel{\vartriangleleft} z$.
• Figure 2: Turning a bicotree into a clean bicotree, when the root is of type O. Color $1$ is red, color $2$ is yellow. Subtrees with monochromatic set of leaves are colored red or yellow; those with both colors are represented with a rising tiling pattern.
• Figure 4: The amalgam $\mathop{\rm Amalg}(V,(\mathbf T_i),(\mathbf T_{i,j}))$. The green parts are the grounds of the structures $\mathbf T_i$ and $\mathbf T_{i,j}$. The ground of the amalgam is the set $V$ (on the dashed circle).
• Figure 5: The coupling of a poset and a graph (on the left, with cover graph in black and graph edges in red) is transduced into a poset (on the right). The adjacency of $u$ and $v$ on the left is reflected by the covers $u_2\prec v_3$ and $v_2\prec u_3$.
• Figure 6: We consider a clean model $\mathbf M$ of a graph $G$. (Hence, $G= \hbox{$\mathsf{SBuild}$}_{n,h}(\mathbf M)$ and $\mathbf M\in\mathop{\mathrm{Sparse}}\nolimits_{n,h}(G)$.) Let $W\supseteq \bigcup_{v\in X}F(u)$. Then, applying $\mathop{\mathrm{Sparse}}\nolimits_{n,3h}$ to $G[W]$ is enough to recover the restriction $\mathbf M\langle X\rangle$ of $\mathbf M$.

Theorems & Definitions (39)

• Conjecture 1: RW_SODA
• Theorem 1
• Theorem 1
• Theorem 1
• Conjecture 1
• Theorem 1
• Conjecture 1
• Theorem 2: BS1985monadic
• Theorem 3: Gurski2000
• Theorem 4: SODA_msrwmsrw