2-Switch-Degree Classification

• 2-Switch-Degree Classification is a framework that stratifies graphs by analyzing 2-switch operations which preserve vertex degree sequences.
• It utilizes combinatorial formulas linking induced subgraph counts and degree invariants, facilitating rapid mixing analysis in random graph sampling.
• The classification impacts structural graph theory by connecting stability notions, extremal properties, and isomorphism changes through 2-switch dynamics.

The 2-switch-degree classification is a framework for analyzing and stratifying degree sequences and graph realizations according to the action of 2-switch operations. At its core, the 2-switch-degree of a graph $G$ is defined as the degree of $G$ in the realization graph $\mathcal{G}(s)$, where $s$ is the degree sequence of $G$. The realization graph connects graphs with the same degree sequence via 2-switches—local edge modifications that preserve vertex degrees. This classification paradigm is pivotal in structural graph theory, Markov chain mixing analysis for random graph sampling, extremal combinatorics, and spectral graph theory, and it has been formalized with connections to number theory in the context of split graphs.

1. The 2-Switch and Realization Graphs

A 2-switch (or simple switching) is an edge transformation on four distinct vertices $a, b, c, d$ of $G$ such that $ab, cd \in E(G)$ and $ac, bd \notin E(G)$. The operation removes $ab, cd$ and inserts $ac, bd$, thus preserving the degree sequence. The realization graph $\mathcal{G}(s)$ for a degree sequence $s$ has as vertices all labeled realizations of $s$, with an edge between two graphs if one can be obtained from the other by a single 2-switch.

The 2-switch-degree of $G$, denoted $d_2(G)$, is:

$$d_2(G) = |\{H \in V(\mathcal{G}(s)) : \{G, H\} \in E(\mathcal{G}(s))\}| .$$

Alternatively, $d_2(G)$ is the total number of active 2-switches, i.e., unordered pairs of disjoint edges for which the corresponding 2-switch can be performed (Pastine et al., 28 Nov 2025).

2. Combinatorial Formulas and Counting

The number of active 2-switches in a graph $G$ is determined via induced subgraphs of 4 vertices. Let $Q_G(X)$ denote the set of induced subgraphs of $G$ isomorphic to $X$, for $X \in \{P_4, C_4, 2K_2\}$:

$d_2(G) = 2|Q_G(2K_2)| + 2|Q_G(C_4)| + |Q_G(P_4)| .$

In split graphs, $C_4$ or $2K_2$ do not appear, hence $d_2(G) = |Q_G(P_4)|$ for such graphs. Further decomposition enables expressions in terms of degree-sequence invariants, triangle counts $k_3(G)$, 4-clique counts $k_4(G)$, and the second Zagreb index $\zeta_2(G)$:

$$d_2(G) = 2\, dpe(s) + 2\, c_4(G) - p_4(G) - 4\, k_4(G) ,$$

where $dpe(s)$ counts disjoint edge-pairs, and similarly, $d_2(G) + \zeta_2(G) = m^2$ holds for graphs with girth at least $5$ (Pastine et al., 28 Nov 2025).

3. Classification of Degree Sequences via 2-Switch Mixing

Degree sequence families are classified by their behavior under the switch (2-switch) Markov chain and the corresponding mixing properties.

• Strong stability and P-stability: A family of sequences is P-stable if small $\ell_1$ perturbations cause only polynomial changes in the number of realizations; strong stability relates to bounded switch-distances in an auxiliary chain. Both notions imply rapid mixing of the 2-switch Markov chain (Amanatidis et al., 2018, Erdős et al., 2019, Gao et al., 2020).
• $k$-stability, notably 8-stability: If for every degree sequence $d'$ within $\|d' - d\|_1 \leq k$, $|\mathcal{G}(d')| \leq M(d)^a |\mathcal{G}(d)|$, $d$ is $(k, a)$-stable. 2-stability coincides with P-stability; 8-stability strictly implies rapid mixing with explicit polynomial bounds (Gao et al., 2020).
• Critical moment and mixing threshold: For random graph sampling, if $\sum d_i^2 = O(n)$ and $\max d_i = o(n^{1/2})$, the law of switched configuration model ($S(n, d)$) is $o(1)$-close in total variation to the uniform distribution. If only $\sum d_i^2 = O(n)$, contiguity (but not total-variation closeness) holds, segmenting degree sequences into "uniform-mixing" and "contiguous-mixing" classes (Janson, 2019).

Explicit sufficient conditions for stability involve:

• $M(d) > 2\, J(d) + 18\, A(d) + 56$ for 8-stability.
• Special classes, e.g., power-law distributions with exponent $y > 2$, meet these for all large $n$ (Gao et al., 2020, Greenhill et al., 2017).

4. Structural and Extremal Classification

Threshold graphs are the only graphs with $d_2(G) = 0$, i.e., no active 2-switches. Upper bounds for $d_2(G)$ are attained for matchings plus isolated vertices. For split graphs, the 2-switch-degree is additive under Tyshkevich composition: $d_2(S \circ G) = d_2(S) + d_2(G)$. Every tree with fixed degree sequence $s$ has $f$-degree $deg_f(T) = dpe(s)$, and for paths $P_n$, $d_2(P_n) = (n-3)^2$ for $n \geq 3$ (Pastine et al., 28 Nov 2025). Unicyclic graphs admit closed formulas parametrized by cycle length.

For split graphs, fine classification is achieved using the factor graph $\Phi(S)$, a multigraph encoding edge multiplicities between independent vertices according to induced $P_4$ counts. The spectrum of 2-switch-degrees is mapped via enumeration over connected multigraphs with prescribed edge-sums. The $\Delta$-property reveals number-theoretic constraints for uniform triangles in $\Phi(S)$, linking structural graph theory to divisor functions and beyond (Schvöllner, 17 Jul 2025).

5. Isomorphism Classes, Unigraphs, and Matroid Connections

Classification can be extended to the impact of 2-switches on graph isomorphism types. Barrus (Barrus, 2011) proves that the only ways a single 2-switch may change isomorphism class occur in four canonical configurations on six vertices, with modules and “symbiotic overlays” blocking such changes. Matrogenic graphs—those where alternating 4-cycles form matroid circuits—are precisely those for which every 2-switch realizes a graph automorphism swapping pairs of vertices in the alternating 4-cycle. The largest hereditary family of unigraphs avoids precisely the forbidden configurations generated by isomorphism-changing 2-switches.

6. Directed Graphs and Switch-Irreducibility

For directed graphs, the 2-switch and its role in the Markov chain is more nuanced. Not all digraphical sequences admit irreducible switch chains: C*-anchored sequences partition realizations into disconnected components, necessitating 3-cycle reorientations for uniform sampling (0912.3834). Outside this class, 2-switches alone suffice for mixing and enumeration. Explicit tests based on slack sequences and conjugate orderings detect anchored triples efficiently.

For directed $k$-stable sequences, analogous mixing bounds hold as in the undirected case. Sufficient conditions involve the total in-degree and out-degree, maxima, and associated critical sums (Gao et al., 2020, Greenhill et al., 2017).

7. Applications, Examples, and Open Directions

Applications span:

• Markov chain Monte Carlo sampling of random graphs with prescribed degrees, leveraging rapid mixing results.
• Spectral theory: ordering trees of fixed degree sequence by spectral radius via 2-switch transitions that strictly decrease the index (Oliveira et al., 2020).
• Classification and enumeration of split prime graphs up to prescribed 2-switch-degree, deploying factor graph techniques and Tyshkevich decomposition (Schvöllner, 17 Jul 2025).
• Diameter-2 graphs: equivalence under degree-restricted 2-switches matches invariance of 2-neighborhood degree lists, enabling robust algorithmic transformation and classification (Benakli et al., 2019).
• Pseudoforests and unicyclic graphs: finite sequences of restricted 2-switches suffice for connectivity within realization space, with polynomial-time algorithms (Jaume et al., 2021).

Open directions include:

• Characterizing degree sequences whose realization graph is complete under 2-switches.
• Fine bounds on $d_2(G)$ under additional combinatorial constraints.
• Complexity analysis for counting $k_4(G)$ and higher-order structures appearing in closed formulas.
• Full articulation of number-theoretic implications of the $\Delta$-property as revealed by factor graphs of split primitives.

The 2-switch-degree classification thus synthesizes local graph operations, global realization graph structure, mixing theory, combinatorial invariants, and algebraic decomposition, providing a robust taxonomic and analytic apparatus for the study of degree sequences and their realizations in both undirected and directed settings.