Star-Based Separators in Graph Theory

• Star-based separators are combinatorial tools that partition graphs into balanced components using disjoint star subgraphs (K₁,k).
• They are constructed via fragmentation, planar contact graphs, and applying planar separator theorems to efficiently decompose intersection graphs.
• Applications include optimizing distance oracles, product structure decompositions, and improving algorithms where traditional vertex separators fall short.

Star-based separators are a combinatorial tool in structural and algorithmic graph theory, providing a framework for partitioning graphs using collections of stars—subgraphs isomorphic to $K_{1,k}$—to achieve balanced decomposition. Star-based separators generalize the classical notion of vertex separators and arise naturally in the analysis of intersection graphs, product structures of certain hereditary classes, and the construction of efficient data structures such as distance oracles. They are particularly impactful in cases where traditional vertex or clique separators are either too large or nonexistent, as exemplified by segment intersection graphs and certain hereditary classes with strongly sublinear separators.

1. Definitions and Fundamental Notions

A balanced separator in a graph $H$ is a subset $S \subseteq V(H)$ such that each connected component of $H - S$ contains at most a fixed fraction (typically $\leq 2/3$) of the total vertices. For a class $\mathcal{G}$ and an $n$-vertex $H \in \mathcal{G}$, let $\mathrm{sep}(H)$ denote the minimum size of a balanced separator. $\mathcal{G}$ admits \textbf{strongly sublinear separators} if there exist constants $\epsilon > 0$, $c > 0$ such that every $H \in \mathcal{G}$ with $n$ vertices satisfies

$\mathrm{sep}(H) \leq c\,n^{1-\epsilon}\,.$

Star-based separators strengthen this paradigm by requiring the separator to be a (vertex-)disjoint union of stars $K_{1,k}$, where each star consists of a center vertex and a subset of its neighbors. The removal of all stars in the separator must ensure the balance property.

Further, for intersection graphs derived from geometric objects, traditional separators may be insufficient due to the existence of large bicliques. Thus, biclique-based separators (collections of bicliques with disjoint vertex sets) and especially star-based separators become central in such settings.

2. Star-Based Separators in Intersection Graphs

Consider a set $V$ of $n$ \emph{pseudo-segments} in the plane—Jordan arcs intersecting at most once per pair, and only by crossing. For $c > 0$, a $c$-colored set of pseudo-segments is a partition $V=V_1 \cup \dots \cup V_c$ such that each $V_i$ consists of pairwise-disjoint curves.

The intersection graph $[V]$ has nodes for pseudo-segments and edges between intersecting pairs. $[V]$ may contain arbitrarily large bicliques, precluding small clique-based separators. However, for constant $c$, every such intersection graph admits a star-based separator of size $O(\sqrt{n})$: there exists a collection of $O(\sqrt{n})$ disjoint stars whose removal partitions $[V]$ into components, each of size at most $2n/3$ (Berg et al., 7 Nov 2025).

This result generalizes to intersection graphs of $c$-oriented polygons, after suitable transformation of each polygon into a collection of oriented sides and containment segments.

3. Algorithmic Construction and Structural Theorems

The construction of star-based separators for $c$-colored pseudo-segments or polygons relies on three main steps:

1. Fragmentation and Contact Graph Construction: Each segment is partitioned at intersections into fragments; interior fragments are selected to ensure the \emph{contact graph} among active fragments is planar.
2. Application of the Planar Separator Theorem: The planar contact graph allows the application of Lipton–Tarjan's theorem, yielding a separator of $O(\sqrt{n})$ fragments.
3. Lifting to Stars in the Intersection Graph: Each separator fragment corresponds to up to three stars in the original intersection graph, ensuring the separator's size remains $O(\sqrt{n})$.

The algorithm executes in $O(n\log n)$ time for $c$-oriented segments and for $c$-oriented polygons, since fragmentation, contact graph construction, and planar separation admit plane-sweep and efficient data structure implementations. Memory usage is $O(n)$ for all representations and outputs.

4. Product Structure and Tree-Depth Decompositions

A distinct but deeply related perspective is provided by product structure theorems for hereditary graph classes admitting strongly sublinear separators (Dvořák et al., 2022). The key result: for such classes $\mathcal{G}$ with $\mathrm{sep}(G) \leq c\,n^{1-\epsilon}$, every $n$-vertex $H \in \mathcal{G}$ is a subgraph of a strong product $S \boxtimes K_m$, where $S$ has bounded tree-depth $\mathrm{td}(S)\leq t$ and

$$m \leq \frac{c\,2^\epsilon}{2^\epsilon - 1} n^{1 - \epsilon + \delta},$$

for any fixed $0 < \delta < \epsilon$ and some $t = t(\epsilon, \delta)$. For the strong product, vertices $(a, b)$ and $(a', b')$ are adjacent if either $a = a'$ and $bb' \in E(K_m)$, $b = b'$ and $aa' \in E(S)$, or both $aa' \in E(S)$ and $bb' \in E(K_m)$.

Allowing $\delta = 0$ incurs a slowly growing depth-bound, specifically $\mathrm{td}(S) = O(\log\log n)$: $m \leq \frac{2c}{2^\epsilon - 1} n^{1 - \epsilon}.$

The exponent gap $\delta$ and $\mathrm{td}(S)$ bounds are shown optimal using isoperimetric inequalities for grid graphs, which enforce lower bounds on possible separator sizes and depth in such decompositions.

For graphs of bounded treewidth $k$, every $n$-vertex graph admits a star-based decomposition with $\mathrm{td}(S) \leq t$ and $m \leq (k+1)^{1-1/t} n^{1/t}$, which is also optimal. This substantiates the breadth of star-based separators as a unifying lens for graph decomposition.

5. Applications to Distance Oracles and Graph Algorithms

Recursive star-based separators enable the construction of efficient almost-exact distance oracles for intersection graphs derived from $c$-colored pseudo-segments and $c$-oriented polygons (Berg et al., 7 Nov 2025). The approach is as follows:

• Preprocessing: Recursively apply star-based separators, dividing the graph into progressively smaller components. At each recursion level, distances from each vertex to every separator center are computed and stored.
• Storage: Across all levels, total storage is $O(n\sqrt{n})$.
• Queries: To find the distance between $s$ and $t$, identify the highest separator separating $s$ and $t$, then take the minimum $d(s,c) + d(c,t)$ over all separator centers $c$. Each such query runs in $O(\sqrt{n})$ time. The result is always within an additive error of at most 2 from the true hop-distance.

This oracle provides the first subquadratic-space, sublinear-query, and additive-bounded structure for such intersection graphs. The construction and query algorithms exploit the recursive balance and locality properties imparted by the star-based separators.

6. Optimality and Limitations

Isoperimetric inequalities for grid graphs and similar structures demonstrate that the scaling of separator size in star-based decompositions is asymptotically tight. For $d$-dimensional grid graphs $G^d_n$, any star-partition must have width $\Omega(n^{d/(d+1)})$, and similar lower bounds exist for bounded treewidth graphs with respect to the parameter $t$ in depth-bounded decompositions.

A central open question concerns attaining exponent gap $0$ \textit{and} bounded tree-width in such product decompositions. Specifically, for hereditary classes with $O(n^{1-\epsilon})$ separators, it is asked whether every $n$-vertex graph embeds in $S \boxtimes K_{O(n^{1-\epsilon})}$ with $\operatorname{tw}(S)\leq c$ for some constant $c$. This is currently known for minor-closed classes, but remains open in general (Dvořák et al., 2022).

7. Representative Examples and Broader Impact

Star-based separators have demonstrable utility across several canonical families:

Graph Class	Separator/Decomposition Size	Structural Notes
Planar graphs	$O(n^{2/3})$ stars	Follows from $O(n^{1/2})$ separator bound
Proper minor-free and surface-embeddable classes	$O(n^{1/2})$ stars	Decomposition with $\operatorname{tw}(S)=O(1)$
$d$-dimensional grids	$\Theta(n^{d/(d+1)})$ stars	Lower bound matches by isoperimetry
Bounded treewidth $k$	$O(n^{1/k})$ stars	Achievable with $\operatorname{td}(S)=O(1)$

The star-based separator viewpoint enables a unified framing: any such graph $G$ is a subgraph of a star (or low-depth forest) blown up by a small complete graph. This insight supports algorithmic metatheorems, improved data structures, and clarifies combinatorial limitations in graph theory.