Strong Separation Number in Graph Theory

• The strong separation number is a graph parameter that quantifies the minimal size of a subgraph family required to uniquely distinguish every pair of edges.
• It exhibits explicit bounds, approaching (1+o(1))n for complete graphs and employing combinatorial constructions for sparse and bipartite graphs.
• Extensions to subdivisions and connections with separation dimensions reveal broader applications in extremal combinatorics and graph decompositions.

The strong separation number of a graph quantifies the minimal size of a family of subgraphs—most structurally, systems of paths or prescribed subdivisions—that are able to distinguish all pairs of edges in a robust combinatorial sense. This parameter governs structural decompositions relevant to extremal combinatorics, separating systems, and combinatorial search, and is centrally linked to the design of sparse representations with strong pairwise discrimination among the graph's edges.

1. Formal Definition and Core Properties

Given a graph $G=(V,E)$, a family $\mathcal{P}$ of subgraphs (notably, simple paths or subdivisions of a fixed graph $H$) is called a strongly separating system if, for every pair of distinct edges $e,f\in E$, there exist subgraphs $P_e,P_f\in\mathcal{P}$ such that $e \in E(P_e), f \notin E(P_e)$, and $f \in E(P_f), e \notin E(P_f)$. The strong separation number, denoted by $\sigma(G)$ (for paths) or $\tau(G,H)$ (for subdivisions of $H$), is the minimal cardinality of such a strongly separating system.

In the path context, this property is equivalently characterized as follows: for $S(e)$ the set of paths in $\mathcal{P}$ containing $e$, it is required that for any $e\neq f$, neither $S(e)\subseteq S(f)$ nor $S(f)\subseteq S(e)$. This notion directly stipulates that no two edges are indistinguishable via the traces they leave across all paths in the system (Fernandes et al., 2023, Fernandes et al., 16 Nov 2025).

2. Extremal Results for Complete and Dense Graphs

For complete graphs $K_n$, the strong separation number for paths, $\sigma_s(K_n)$, satisfies $\sigma_s(K_n) = (1+o(1))n$. The lower bound is realized by elementary counting: each of the ${n \choose 2}$ edges must be separated from each other, and each path can cover at most $n-1$ edges, while each edge must appear in at least two different paths to be separable from all others. The matching upper bound arises from sophisticated constructions employing random orientations, hypergraph covering arguments, and robust decompositions into low-degree structures, ultimately patched into Hamilton-like paths (Fernandes et al., 2023).

For regular graphs of degree $\alpha n$, the asymptotic formula generalizes as $\sigma_s(G) = (\sqrt{3\alpha+1} - 1 + o(1))n$ for large $n$, contingent on robust connectivity assumptions. The tightness up to the $o(n)$ term is established by precise combinatorial arguments analyzing how often edges can coexist across the chosen subgraph system.

3. Linear Upper and Lower Bounds for Sparse Graphs

For the broad class of $2$-degenerate graphs, which encompasses all graphs whose subgraphs have minimum degree at most 2, the strong separation number for paths is bounded above by the number of vertices: $\sigma(G) \leq n$. This is achieved by an explicit inductive scheme constructing $n$ paths for a connected graph of order $n$, with the additional property that each edge is in exactly two paths and each vertex is an endpoint of exactly two paths. This bound propagates to wider graph classes via decompositions: for subcubic graphs ($\Delta\leq3$), planar graphs ($\sigma(G)\leq 2n$), and planar bipartite graphs ($\sigma(G)\leq \frac{3}{2} n$) (Fernandes et al., 16 Nov 2025).

For highly asymmetric bipartite graphs $K_{a,b}$, $\sigma(K_{a,b}) = b$ when $a < b/2$, as explicit graceful label-based constructions achieve the bound, and the lower bound is immediate by maximal degree considerations. In the more balanced regime $b/2 \leq a \leq b$, sharp lower bounds are derived from edge-multiplicity counting arguments, yielding $\sigma(K_{a,b}) \geq (\sqrt{6(b/a)+4} - 2) a$, which is tight for the boundary cases (Fernandes et al., 16 Nov 2025).

4. Strong Separation by Subdivisions and General Families

A generalization considers separating systems consisting of subdivisions of a given graph $H$, rather than merely paths. For two graphs $G$ and $H$, the strong separation number by subdivisions is denoted $\tau(G, H)$: the smallest number of $H$-subdivisions and single edges required to strongly separate all edge pairs in $G$. The main theorem (Kontogeorgiou et al., 16 Jun 2025) establishes that for every fixed $H$, there is an absolute constant $C_H$ such that $\tau(G, H) \leq C_H |V(G)|$ for any $G$. This resolves the longstanding Botler–Naia conjecture and demonstrates that allowing arbitrarily large $H$-subdivisions suffices to beat the trivial quadratic bound for general strong separation systems of edges.

For 3-connected graphs, the bound is made explicit: for every $H$ and 3-connected $G$, there is a strongly separating system of subdivisions of size $O(|H|^2 n)$, via a combination of Tutte decompositions, 3-connected torso analysis, and iterative attachment of cycles for separation (Kontogeorgiou et al., 16 Jun 2025).

5. List-Separation as a Related Graph Parameter

A related but formally distinct invariant is the separation number in list coloring with separation, denoted $\mathrm{sep}(G, a, b)$, which is the largest integer $c$ such that for any assignment of lists of size $a$ to each vertex (with pairwise intersection on edges at most $c$), one can choose subsets of size $b$ at each vertex that assign disjoint color sets to adjacent vertices. While not a separation number for edge-detection, this parameter is deeply connected to intersecting set systems and extremal combinatorics (Godin et al., 2022).

Table: Summary of Key Strong Separation Numbers

Graph Family	Strong Separation Number	Tightness/Method
$K_n$ (complete)	$(1+o(1))n$	tight, combinatorial (Fernandes et al., 2023)
$K_{a,b}$, $a<b/2$	$b$	achieved by construction (Fernandes et al., 16 Nov 2025)
$2$-degenerate	$\leq n$	tight, inductive (Fernandes et al., 16 Nov 2025)
$\alpha n$-regular	$(\sqrt{3\alpha+1} - 1 + o(1))n$	extremal argument (Fernandes et al., 2023)
$G$, Sub(H)	$O(|H|^2 n)$	general, explicit (Kontogeorgiou et al., 16 Jun 2025)

6. Open Directions and Future Work

Several regimes remain open for precise determination of the strong separation number, especially for balanced bipartite graphs with $b/2 < a < b$, various highly symmetric structures (e.g., hypercubes), and for extensions to separated systems over general subgraph families. Reducing the current quadratic dependence on $|H|$ in strong separation by subdivisions, further tightening constant factors, and characterizing minimal separating systems in additional graph classes constitute active branches. The connection between strong separation and weak separation parameters, as well as applications to search and detection problems in combinatorics, remain significant areas of exploration (Kontogeorgiou et al., 16 Jun 2025).

7. Connections to Separation Dimension and Related Invariants

The strong separation number is distinct from, but related to, the separation dimension $\pi(G)$ (Basavaraju et al., 2014), which seeks the minimal number of axis-parallel hyperplane directions or permutations needed to separate every pair of disjoint edges in a graph via geometric or order-based methods. While $\pi(G)$ is $\Theta(\log n)$ for $K_n$, the strong separation number for edges (as defined above) has fundamentally different growth rates, especially in dense and sparse regimes. For instance, for 2-degenerate graphs, $\pi(G)=O(\log\log n)$, contrasting with the linear upper bound for the strong separation number. This distinction underlines the stronger combinatorial requirements imposed by the strong separation paradigm compared to order- or axis-oriented analogues.

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References:

• (Fernandes et al., 2023) Separating path systems in complete graphs
• (Fernandes et al., 16 Nov 2025) Separating path systems for cubic graphs and for complete bipartite graphs
• (Kontogeorgiou et al., 16 Jun 2025) Separating edges by linearly many subdivisions
• (Godin et al., 2022) On List Coloring with Separation of the Complete Graph and Set System Intersections
• (Basavaraju et al., 2014) Separation dimension of sparse graphs