λ-Linked Partition: Structure & Algorithms

• λ-linked partition is a combinatorial structure that splits a graph into cliques or Hamiltonian-λ-linked parts to guarantee robust connectivity.
• It leverages structural properties of geometric intersection graphs, ensuring bounded treewidth and low contraction graph degree.
• The partition framework supports efficient, ETH-tight algorithms for problems like Hamiltonian Cycle, Long Path, and robust routing.

A $λ$-linked partition is a combinatorial structure introduced to facilitate the design of robust subexponential algorithms for connectivity problems in geometric intersection graphs—graphs in which vertices represent geometric objects (typically “similarly sized fat objects” in $\mathbb{R}^d$) and edges indicate intersection. A $λ$-linked partition decomposes the vertex set into parts, each inducing either a clique or a highly connected “Hamiltonian-λ-linked” subgraph. This enables key routing and coverage properties essential to robust algorithm design. The concept was formalized and algorithmically realized in the context of geometric intersection graphs to achieve efficient, ETH-tight algorithms for problems such as Hamiltonian Cycle, Hamiltonian Path, and parameterized Long Path, all without requiring explicit geometric information (Marin et al., 3 Dec 2025).

1. Formal Definition of a $λ$-Linked Partition

Given a simple undirected graph $G$ and integer parameter $\lambda \ge 1$, a subgraph $H \subseteq G$ is called Hamiltonian-λ-linked if $|V(H)| \ge 2\lambda$ and, for every collection of $\lambda$ ordered pairs $(s_1, t_1), \ldots, (s_\lambda, t_\lambda)$ of disjoint vertices, there exist $\mathbb{R}^d$0 vertex-disjoint paths $\mathbb{R}^d$1 in $\mathbb{R}^d$2 such that each $\mathbb{R}^d$3 connects $\mathbb{R}^d$4 to $\mathbb{R}^d$5 and the union of the $\mathbb{R}^d$6 covers all of $\mathbb{R}^d$7.

A $\mathbb{R}^d$8-linked partition of $\mathbb{R}^d$9 is a partition $λ$0 of $λ$1 such that, for each part $λ$2, one of the following holds:

• $λ$3 is a clique,
• $λ$4 is Hamiltonian-λ-linked.

Thus, a $λ$5-linked partition produces a decomposition where every part is either trivial (clique) or admits strong connectivity facilitating simultaneous linkage of up to λ disjoint terminal pairs in a Hamiltonian fashion (Marin et al., 3 Dec 2025).

2. Existence and Structural Theorems

On intersection graphs of similarly sized β-fat objects in $λ$6, a foundational structural result guarantees the existence of $λ$7-linked partitions with desirable global properties. Specifically, for fixed $λ$8 and $λ$9, there are constants $λ$0, $λ$1, and $λ$2 (with $λ$3), such that any intersection graph $λ$4 of this type admits a $λ$5-linked partition $λ$6 where:

• The contraction graph $λ$7 (whose nodes are the parts $λ$8 with edges corresponding to inter-part adjacencies in $λ$9) has maximum degree at most $G$0.
• $G$1 has treewidth $G$2, where $G$3.

This result is built on recursively refining a constant-bounded clique-cover partition (the initial “κ-partition” from de Berg et al.) using small separator trees. Subpieces are classified as either cliques or highly connected components. The crucial lemma—building on results of Fomin et al.—states that high vertex-connectivity and small independence number ensure Hamiltonian-λ-linkedness for sufficient $G$4. This yields parts suitable for robust algorithmic applications (Marin et al., 3 Dec 2025).

3. Polynomial-Time Computation

A $G$5-linked partition of $G$6 can be computed in polynomial time through the following process:

1. Compute an initial κ-partition using the polynomial-time algorithm of de Berg et al.
2. For each part $G$7 in the κ-partition, recursively build a separator tree:
   • If no small ($G$8-sized) vertex separator exists, recognize $G$9 as “g-connected.”
   • Otherwise, decompose $\lambda \ge 1$0 along such a separator, recurse, and gather all subpieces.
3. The leaves of the separator tree are g-connected components; internal nodes are separators of bounded size. Union them as needed and brute-force any necessary clique partitions of the separators (bounded size).
4. Refine all parts using this approach, yielding at most $\lambda \ge 1$1 subparts per original piece.
5. Set $\lambda \ge 1$2 large enough to meet the connectivity threshold ensuring each non-clique piece is Hamiltonian-λ-linked.

All constants ($\lambda \ge 1$3) are independent of $\lambda \ge 1$4, and all steps are polynomial in $\lambda \ge 1$5 (Marin et al., 3 Dec 2025).

4. Algorithmic Applications and Impact

The primary motivation for $\lambda \ge 1$6-linked partitions is efficient, robust algorithm design in geometric intersection graphs, especially for connectivity and routing problems. Notably:

• Hamiltonian Cycle and Path: Compute a $\lambda \ge 1$7-linked partition and a low-treewidth decomposition of the contraction graph. Augment each part into a clique (forming $\lambda \ge 1$8). Apply sparsification lemmas to bound inter-part adjacencies, then use treewidth-based dynamic programming (DP) to solve for Hamiltonian structures in time $\lambda \ge 1$9. If a solution uses only limited blue-edges (crossing parts), the properties of the λ-linked partition enable reconstructing a true Hamiltonian cycle/path in $H \subseteq G$0.
• Long Path (parameterized by $H \subseteq G$1): Again use $H \subseteq G$2-linked partitions; contract unmarked vertices via marking lemmas, creating a weighted graph with preserved geometric and contraction structure. Apply random low-treewidth pattern covering to retain all possible $H \subseteq G$3-size sets with high probability and solve the problem with DP. The overall expected time is $H \subseteq G$4.

Critically, these methods do not require geometric embedding or coordinate knowledge. All computations and correctness rely solely on the abstract graph and the invariants guaranteed by the $H \subseteq G$5-linked partition (Marin et al., 3 Dec 2025).

5. Key Lemmas and Theoretical Foundations

The effectiveness of the $H \subseteq G$6-linked partition method arises from several structural and algorithmic lemmata:

• High Connectivity $H \subseteq G$7 Hamiltonian-λ-Linked: Any $H \subseteq G$8-vertex graph with vertex connectivity at least $H \subseteq G$9 and independence number at most $|V(H)| \ge 2\lambda$0 is Hamiltonian-λ-linked for any $|V(H)| \ge 2\lambda$1.
• Bounded Clique-Cover and Separator Hierarchy: Initial partitions with bounded clique number can support the construction of separator trees yielding subparts that respect the Hamiltonian-λ-linked criteria.
• Treewidth and Bounded Degree of the Contraction: The contraction of the $|V(H)| \ge 2\lambda$2-linked partition retains both bounded degree and sublinear treewidth, facilitating dynamic programming and advanced covering techniques.

A plausible implication is that this partition paradigm extends the toolkit for exploiting both local and global structure in complex graph classes without the need for geometric representation (Marin et al., 3 Dec 2025).

6. Generalizations and Broader Applicability

The concept of a $|V(H)| \ge 2\lambda$3-linked partition abstracts the process of clustering vertices into highly connected “blobs” (either cliques or supporting robust multi-terminal linkage) whose contractions yield graphs amenable to recursive and DP algorithms. This suggests broader applicability:

• Other Connectivity-Sensitive Problems: Extensions to Steiner Tree, Steiner Forest, $|V(H)| \ge 2\lambda$4-disjoint paths, edge-disjoint paths, cycle packing, and related variants in intersection graphs.
• Alternative Intersection Graph Classes: Any class permitting constant-bounded clique covers with low-treewidth contraction (e.g., unit-ball graphs, intersection graphs of other “fat” objects).
• Potential in Directed or Weighted Settings: The partitioning approach and separator techniques are plausible to generalize wherever small separator hierarchies are structurally available.

The $|V(H)| \ge 2\lambda$5-linked partition framework thus unifies local connectivity strength with coarse global structure, forming a robust and flexible basis for algorithmic design in geometric intersection graphs and beyond (Marin et al., 3 Dec 2025).