Biclique-Based Separators: A Geometric Approach

• Biclique-based separators are collections of vertex-disjoint complete bipartite subgraphs that, when removed, partition a graph into components of bounded size.
• They leverage the structural properties of geometric intersection graphs (from pseudo-segments and c-oriented polygons) and can be computed in O(n log n) time.
• Their application in constructing distance oracles leads to subquadratic space and sublinear query time with a controlled additive error, enhancing algorithmic performance.

A biclique-based separator in a graph-theoretical context refers to a small collection of vertex-disjoint bicliques—complete bipartite subgraphs—whose removal decomposes the graph into subgraphs of bounded size. The concept has emerged as a response to challenges in segment and string intersection graphs, which can contain large cliques and bicliques, rendering classical separator techniques based on individual vertices or cliques ineffective. Star-based separators are a particularly tractable subclass wherein each biclique is a star (a $K_{1,r}$, that is, a center linked to $r$ leaves). The recent introduction and systematic analysis of biclique-based and star-based separators has enabled algorithmic advances in geometric intersection graphs, especially for $c$-colored pseudo-segments and $c$-oriented polygons (Berg et al., 7 Nov 2025).

1. Formal Definitions and Model

The paper of biclique-based separators is rooted in intersection graphs derived from geometric objects, particularly pseudo-segments and polygons constrained by coloring or orientation.

• Intersection Graph of Pseudo-Segments: Given a finite set $V$ of Jordan arcs (pseudo-segments) in the plane such that any two are either disjoint or cross exactly once, the intersection graph $[V]$ has vertex set $V$ and an edge $\{u,v\}$ if $u \cap v \neq \emptyset$.
• $c$-Colored and $c$-Oriented Sets: A set $V$ is $c$-colored if partitioned into $c$ classes ($V=V_1 \cup ... \cup V_c$) with pairwise disjoint pseudo-segments within each class. For segments, $c$-oriented means each segment aligns with one of $c$ pre-fixed directions, making the set $c$-colored with colors by orientation.
• Biclique-Based and Star-Based Separator: In a graph $G=(V,E)$, a collection $\mathcal{S} = \{B_1, ..., B_t\}$ of pairwise vertex-disjoint subgraphs forms a balanced biclique-based separator if each $B_j$ is a complete bipartite subgraph (a biclique) and their removal splits $G$ into components of size at most $2|V|/3$. If each $B_j$ is a star $K_{1,r}$, it is a star-based separator. The size of $\mathcal{S}$ is $t$.

These definitions formalize the separator property with bicliques as atomic units, mirroring classic separator objectives but accommodating dense local structure typical of segment and string intersection graphs.

2. Structural Theorems and Main Results

The existence and construction of small biclique- or star-based separators in geometric intersection graphs is governed by a central theorem:

• Theorem (Star-Based Separator for $c$-Colored Pseudo-Segments):

For a $c$-colored set $V$ of $n$ pseudo-segments (or non-overlapping $c$-oriented segments) each with nonnegative weights summing to $1$, the intersection graph $G=[V]$ admits a star-based separator $\mathcal{S}$ of size $O(\sqrt{n})$ such that after removing the centers of all stars in $\mathcal{S}$, each component has total weight at most $2/3$. For explicit $c$-oriented segments in general position, such a separator can be found in $O(n\log n)$ time.

A similar result holds for $c$-oriented polygons of constant complexity, where the intersection graph $[\mathcal{P}]$ of $n$ polygons with edges in $c$ directions also admits a star-based separator of size $O(\sqrt{n})$, constructible in $O(n\log n)$ time.

These theorems generalize planar separator results to complex geometric intersection graph families for which classical methods fail, notably when large bicliques preclude the possibility of small vertex- or clique-based separators.

3. Methodological Framework and Proof Techniques

The separator construction for $c$-colored pseudo-segments follows a three-stage methodology:

1. Fragmentation into a Planar Contact Graph:
   • Iterate through color classes. For each, split segments at intersections with existing fragments, labelling sub-arcs as end fragments (containing segment endpoints) or internal fragments (bridging older fragments).
   • End fragments become "active." Internal fragments are grouped by equivalence, activating a single representative if the enclosed quadrilateral is empty of endpoints.
   • The active fragment set $F$ forms a planar contact graph $H=(F, \{\{f, f'\} : f \cap f' \neq \emptyset\})$ with $|F|=O(n\,4^c)=O(n)$ nodes.
2. Weighted Planar Separator:
   • Assign weight $1/n$ to a representative fragment per original segment (others get weight $0$).
   • Apply the planar separator theorem to $H$ to obtain a set $S_H$ of $O(\sqrt{n})$ fragments whose removal yields components of weight at most $2/3$.
3. Lifting to Stars in Intersection Graph:
   • For each separator fragment $f\in S_H$:
     • If $f$ is an end fragment of $s$, include the star centered at $s$.
     • If $f$ bridges fragments from $s_g$ and $s_{g'}$, add stars centered at $s_f$, $s_g$, $s_{g'}$.
   • At most $O(1)$ stars are added per fragment, yielding an $O(\sqrt{n})$-sized separator.
   • Assignments ensure no edges cross between the resulting partitioned sets, and component weights remain bounded.

For $c$-oriented polygons, the method adapts by representing each polygon as a set of edges and auxiliary segments, reducing the problem to the pseudo-segment case without increasing asymptotic size.

4. Algorithmic Applications: Distance Oracles

A principal algorithmic consequence is the design of efficient almost-exact distance oracles for intersection graphs of $c$-colored pseudo-segments. Using the star-based separator of size $O(\sqrt{n})$:

• Distance Oracle Construction:
  • Build a recursive separator tree; at each node, store, for every vertex $v$ and every separator star $S$, the value $\delta(v, S) = \min\{d(v, u): u\in S\}$.
  • Each tree level consumes $O(n\sqrt{n})$ storage with $O(\log n)$ recursion depth.

The resulting data structure achieves:

• $O(n\sqrt{n})$ space and preprocessing time,
• $O(\sqrt{n})$ query time,
• Additive error in hop distance at most $2$. The increased error compared to clique-separator-based oracles (which achieve $+1$) is due to the internal structure of stars as bicliques.

These properties constitute the first subquadratic-space, sublinear-query-time distance oracles for such intersection graph classes, surpassing previous methods which could not support such efficiency with bounded additive error.

5. Limitations and Extensions

The methodological advances carry inherent limitations and open problems:

• The $O(\sqrt{n})$ bound for the size of separators is tight; for instance, grid-like configurations achieve this bound.
• For general string graphs, only weaker separators of size $O(n^{2/3}\log^{2/3}n)$ are possible with existing techniques, and extending results to arbitrary curves (non-fixed $c$) is unresolved.
• Reducing the additive error from $2$ to $1$ for segment graphs remains unachieved.
• Extension to higher-dimensional intersection graphs, such as those formed from boxes or fat objects, is unexplored.
• The potential for biclique- or star-based separators to improve algorithms beyond distance queries (e.g., for dominating set or routing problems) in non-fat geometric graphs is a direction of possible future research.
• Tightening combinatorial bounds for string-graph separators could immediately yield more powerful oracles.

A plausible implication is that progress in bounding the separator size for broader graph classes would have direct practical impact on a range of computational geometry problems, particularly those where classical separator approaches are infeasible due to the presence of large cliques or bicliques.

6. Contextual Significance and Applications

Biclique-based separators, and notably star-based separators, have reshaped the understanding and practical handling of geometric intersection graphs that do not admit traditional separator approaches. By leveraging geometric structure and the interplay of color classes or segment orientations, these separators allow for:

• Decomposition of complex graphs while capturing dense local structure,
• Design of algorithmic tools, such as efficient almost-exact distance oracles with bounded error and subquadratic storage,
• Improved insight into the intersection patterns that govern the computational complexity of geometric algorithms.

These developments illustrate a shift from vertex- or clique-centric separator philosophy to more refined objects in graph partitioning, reflecting the complexity and richness of modern computational geometry and its algorithmic landscape (Berg et al., 7 Nov 2025).