Dynamic Connected Dominant Sets in Graph Theory

• Dynamic CDS is a structure where a vertex subset dominates a graph and induces a connected subgraph, forming the basis for efficient partitioning.
• The algorithm leverages k disjoint CDSs to construct vertex partitions in polynomial time that meet prescribed size and connectivity constraints.
• Applications span network backbone design, distributed processing, and dynamic reoptimization, demonstrating the practical impact of CDS-based methods.

A connected dominating set (CDS) in a graph is a vertex subset D ⊆ V(G) that is both dominating (every vertex not in D has a neighbor in D) and induces a connected subgraph G[D]. The classical Győri–Lovász theorem asserts that any k-connected graph G can be partitioned into k connected subgraphs, each containing a prescribed vertex and having a prescribed size, provided the total matches |V(G)|. Although existentially powerful, this theorem is non-constructive and known algorithms for its partitions run in exponential time even for moderate k. New algorithmic progress leverages dynamic connected dominating sets—specifically, the existence of k disjoint CDSs—to efficiently construct partitions, with consequences for both structural and algorithmic graph theory, particularly in the context of dynamic and high-connectivity networks (Niklanovits et al., 17 Mar 2025).

1. Connected Dominating Sets and the Győri–Lovász Theorem

A connected dominating set D ⊆ V(G), or CDS, is a fundamental structure in both combinatorial optimization and network design. The Győri–Lovász theorem states that any k-connected graph G allows a partition V = V₁ ∪ ⋯ ∪ V_k with |V_i| = n_i, each part V_i connected, and each containing an assigned terminal c_i, for any vector of sizes n_1,...,n_k summing to |V(G)|.

While the original theorem is existential and proof techniques yield algorithms running in exponential time, the key computational barrier is maintaining the prescribed connectivity and domination simultaneously while partitioning according to the given specification.

Recent work demonstrates that if G contains k disjoint CDSs—a much stronger requirement than mere k-connectivity—the desired partition can be computed in polynomial time. This leads to the concept of a CDS partition: a partition of V into k disjoint subsets, each of which is a CDS.

2. Algorithmic Construction of Connected Partitions via CDS

The main algorithmic result is that—given k disjoint CDSs together with prescribed terminals and part sizes—one can efficiently assign vertices to k subsets V₁,...,V_k such that each V_i contains c_i, has size n_i, and G[V_i] is connected. The strategy proceeds as follows:

• CDSs are modeled as vertex-disjoint dominating trees.
• Trees are labeled and sorted according to which terminals they contain.
• When a tree contains multiple terminals, its vertices are split among the corresponding parts while preserving connectivity.
• Iterative routines using auxiliary labeling functions (Vlabel, Tlabel) repeatedly add vertices to parts, updating when a part has fulfilled its size requirement.
• Additions maintain that vertices are included only along adjacencies inherited from the dominating trees, thereby preserving connectivity within each part.

The process is guaranteed to terminate with a feasible partition, and all steps are shown to run in polynomial time (with a worst-case bound of O(n⁵)), assuming that the initial CDS partition is given.

3. Conditions on the Existence of CDS Partitions and Connectivity Thresholds

The existence of the necessary CDS partition (i.e., k disjoint CDSs) is tightly linked to the underlying graph class and its connectivity:

Graph Class	Required Connectivity for CDS Partition of Size k	Efficient Constructive Partition
General graphs	Ω(k * log² n)	Yes (approximate)
Convex bipartite graphs	4k	Yes (approximate)
Biconvex/interval graphs	k	Yes (exact, "true" constructive)

In general, for a Győri–Lovász partition with k parts, the input graph must have connectivity at least Ω(k * log² n) (Niklanovits et al., 17 Mar 2025). This is polylogarithmically higher than the classical statement but admits an efficient construction. In the special cases of biconvex and interval graphs, k-connectivity is sufficient to guarantee a CDS partition of size k and hence a "true" constructive version. For convex bipartite graphs, 4k-connectivity suffices.

These findings rest on prior results such as Censor-Hillel et al., which give CDS partitions in highly connected graphs with size proportional to connectivity modulo polylogarithmic factors.

4. Applications and Implications

Efficient partitioning into k prescribed, connected subgraphs has far-reaching applications:

• Network Backbone Design: Building virtual backbones in wireless or sensor networks requires fault-tolerant partitions; CDSs provide redundancy and coverage.
• Distributed Processing: Assigning connected subgraphs of designated sizes to different computational agents or routers.
• Graph Drawing and Visualization: Large-scale graph layouts often decompose graphs into manageable connected parts.
• Dynamic and Robust Networks: In evolving networks, maintaining CDS partitions enables on-the-fly reconfiguration of communication backbones.

A significant implication is that the presence of k disjoint CDSs can serve as a proxy for vertex connectivity in algorithmic graph theory, enabling efficient partitioning and resource allocation.

5. Dynamic and Reoptimization Scenarios

The modular structure and the vertex-based routines of the CDS-partitioning algorithm indicate strong potential for dynamic settings:

• If vertices or edges are added or removed (as in dynamic graphs), reassignments may be limited to affected regions.
• When a part becomes "full" (reaching its size constraint), the algorithm can restart on the remainder, ensuring continuity.
• Local updates of dominating trees, leveraging the adjacency-based expansions, allow for efficient incremental recomputation.

A plausible implication is that CDS-based partitioning admits efficient incremental algorithms under local topology changes, especially in networks where CDS maintenance is already performed for routing or control.

6. Future Directions and Open Questions

Key avenues of research highlighted by this approach include:

• Trade-offs: Investigating more refined trade-offs between graph connectivity and the size or efficiency of computable CDS partitions. Is there a continuous "interpolation" between the extreme requirements?
• Characterizing Graph Classes: Determining which additional (beyond biconvex and interval graphs) admit k CDSs at mere k-connectivity. For instance, chordal graphs and convex bipartite graphs require greater than k-connectivity for such results.
• Approximation and Parameterized Algorithms: Can approximation algorithms deliver large CDS partitions in polynomial time for lower connectivity graphs? What about parameterizations by treewidth or other structural invariants?
• Dynamic Algorithms: Extending the current framework to fully dynamic networks with robust online maintenance of CDS partitions and associated connected partitions.
• Generalization: Tightening the polylogarithmic gap between the connectivity needed for existential versus constructive versions of the Győri–Lovász theorem.

7. Comparative Context and Broader Impact

This constructive approach, by substituting a CDS partition for vertex connectivity as the driving invariant, advances the algorithmic accessibility of the Győri–Lovász theorem well beyond prior existential or brute-force methods. It achieves polynomial time for a class of graphs previously restricted to exponential constructions and links the combinatorial geometry of CDSs with core partitioning theorems in graph theory (Niklanovits et al., 17 Mar 2025). Furthermore, it establishes a template for approaching other existential combinatorial results through domination and connectivity structures that can be monitored dynamically and updated efficiently.

In summary, CDS-based constructive partitions provide both theoretical unification and practical tractability for graph partitioning tasks in high-connectivity and dynamic scenarios.