On coarse tree decompositions and coarse balanced separators

TL;DR

This paper investigates how coarse graph theory can extend the classic link between treewidth and balanced separators to settings where both tree-decomposition bags and separators are coverable by bounded-radius balls. The authors introduce the notion of a distance-based, coarse separator number and study its relationship with coarse tree decompositions, showing two main directions: (i) under general conditions one can obtain tree decompositions with bags coverable by a logarithmic-factor number of balls per radius, and (ii) when the graph has bounded doubling dimension, the coarse equivalence between small balanced separators and small-width coarse tree decompositions becomes fully valid, including quasi-isometric relations to bounded-degree, tree-partitioned graphs. They develop a distance-graph framework and a quasi-isometry toolkit to transfer properties between graphs of bounded degree and their coarse counterparts, and they deploy a two-tier analysis: a simpler, log-growth bound and a more delicate radii-growth bound controlled by a potential function. They further show that the seemingly scale-dependent coarse statements for r>1 reduce to the r=1 case in unweighted graphs via distance-graph constructions, enabling a unified treatment across scales. Overall, the work extends the foundational connection between separators and tree-like structure to coarse graphs, with implications for coarse algorithms and structural graph theory in spaces of bounded doubling dimension.

Abstract

It is known that there is a linear dependence between the treewidth of a graph and its balanced separator number: the smallest integer $k$ such that for every weighing of the vertices, the graph admits a balanced separator of size at most $k$. We investigate whether this connection can be lifted to the setting of coarse graph theory, where both the bags of the considered tree decompositions and the considered separators should be coverable by a bounded number of bounded-radius balls. As the first result, we prove that if an $n$-vertex graph $G$ admits balanced separators coverable by $k$ balls of radius $r$, then $G$ also admits tree decompositions ${\cal T}_1$ and ${\cal T}_2$ such that: - in ${\cal T}_1$, every bag can be covered by $O(k\log n)$ balls of radius $r$; and - in ${\cal T}_2$, every bag can be covered by $O(k^2\log k)$ balls of radius $r(\log k+\log\log n+O(1))$. As the second result, we show that if we additionally assume that $G$ has doubling dimension at most $m$, then the functional equivalence between the existence of small balanced separators and of tree decompositions of small width can be fully lifted to the coarse setting. Precisely, we prove that for a positive integer $r$ and a graph $G$ of doubling dimension at most $m$, the following conditions are equivalent, with constants $k_1,k_2,k_3,k_4,Δ_3,Δ_4$ depending on each other and on $m$: - $G$ admits balanced separators consisting of $k_1$ balls of radius $r$; - $G$ has a tree decomposition with bags coverable by $k_2$ balls of radius $r$; - $G$ has a tree-partition of maximum degree $\leq Δ_3$ with bags coverable by $k_3$ balls of radius $r$; - $G$ is quasi-isometric to a graph of maximum degree $\leq Δ_4$ and tree-partition width $\leq k_4$.

Figures (2)

• Figure 1: The construction of $T"$ in the proof of \ref{['thm:equivalences']}. Left: The definition of the sets $I$ (red), $C_z$, and $C_y$ for a vertex $y \in V(T)$ (both sets blue). Right: The vertices and edges of $T"$.
• Figure 2: Various sets of balls in the proof of \ref{['thm:tdecomp']}. Blue disks indicate balls from $\mathcal{D}$ and their centers, i.e., the set $O_{\mathcal{D}}$, are shown by blue dots. Balls $\mathcal{D}_W$ are obtained by (possibly) enlarging balls from $\mathcal{D}$, while keeping the same centers; this is not shown in the picture for the sake of clarity. Red disks depict balls from $\mathcal{B}'$. Balls filled with diagonal lines have their centers in $W$, i.e., they belong to $\mathcal{B}'_W$. Filled red disks indicate the set $\widehat{\mathcal{D}}$, i.e., their centers are close to the vertices from $O_{\mathcal{D}}$.

Theorems & Definitions (36)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1: tree-partitionsWood09
• Lemma 2.2: see e.g. HarveyW17
• Lemma 3.1
• proof
• Lemma 3.2
• proof