Vertex-Ordering Characterizations

• Vertex-ordering characterizations are formal methods that classify graphs and digraphs by imposing specific vertex order constraints and forbidden patterns.
• They leverage pattern avoidance and degree-bound criteria, such as perfect elimination orderings in chordal graphs, to guide efficient recognition algorithms.
• These frameworks unify diverse graph properties, enabling practical algorithm design, complexity analysis, and extensions to geometric and multipartite structures.

A vertex-ordering characterization is a formal method that precisely describes graph, digraph, or combinatorial structure classes in terms of existence (or absence) of vertex orderings satisfying specific structural, degree, or pattern constraints. Such characterizations play a central role in recognition algorithms, algorithmic meta-theorems, complexity dichotomies, and combinatorial proofs across extremal graph theory, structural theory, and applied computational mathematics.

1. Fundamentals of Vertex-Ordering Characterizations

Let $G=(V,E)$ denote a graph or digraph. A vertex ordering is a bijection $\pi: V \to \{1,2,\ldots, |V|\}$, often denoted $v_1, v_2, \ldots, v_n$. Vertex-ordering characterizations establish that $G$ belongs to a target class $\mathcal{C}$ if and only if there exists $\pi$ with prescribed ordering properties. These properties may include local constraints (e.g., absence of certain forbidden suborderings/patterns), degree-bounds in specific positions, or global structural consequences for neighborhood or adjacency relations.

A canonical instance is the perfect elimination ordering for chordal graphs, where every vertex's higher-numbered neighbors must induce a clique. More generally, this framework captures transitive orientations (comparability), cocomparability, orderings with matrix/interval/block structure, and combinatorial constraints.

2. Vertex-Ordering Patterns and Forbidden Substructures

A principal form of ordering characterization relies on pattern avoidance: specifying a finite (or sometimes infinite) set $\Pi$ of forbidden induced subgraphs or adjacency patterns that depend on the linear ordering of their vertices. For example, in cocomparability graphs, an ordering must avoid the "umbrella" pattern ($P_4$ in ordering language: $u < v < w$ with $uw \in E$ but $uv, vw \notin E$) (Habib et al., 2017). For interval $r$-graphs, forbidden 3-vertex and 4-vertex patterns are classified in detail and correspond to minimal obstructions for realizing an interval representation in accordance with a vertex ordering (Paul et al., 16 Jan 2026). In circular-arc generalizations, more complex 4-vertex pattern families are required (Paul et al., 13 Mar 2025, Paul et al., 22 Sep 2025).

These forbidden patterns dictate both structural and algorithmic boundaries; their presence or absence provides certificates for membership and guides efficient recognition.

3. Degree-Constrained and Greedy Orderings

Another major thread is degree-bounded orderings in digraphs or undirected graphs. For a digraph $D=(V,A)$ and ordering $\pi$, arcs are partitioned into left-going and right-going arcs with respect to $\pi$. Local quantities such as left-outdegree ($\delta^\ell$) and right-indegree ($\rho^r$) become pivotal (Borsik et al., 5 Sep 2025). Core problems involve the existence of $\pi$ with per-vertex lower and/or upper bounds on these degrees.

Greedy algorithms can construct such orderings efficiently when only lower or upper bounds are imposed, paralleling notions of degeneracy or acyclic orientations. However, when both sides are constrained, or certain exact bounds are required, the problem's complexity sharply increases (to NP-complete or worse).

These degree-bounded orderings underpin foundational results in graph structure (e.g., $k$-degeneracy), algorithmic cycle coverings, arborescence decompositions, and have direct connections to other problems such as rank aggregation and threshold reachability (Borsik et al., 5 Sep 2025).

4. Vertex Ordering and Algorithmic Graph Theory

Vertex ordering is a unifying abstraction in graph algorithms. For instance, in chordal graphs, a perfect elimination ordering not only characterizes the class but also yields fill-minimizing Cholesky factorizations, tractable inference in Gaussian graphical models, and efficient recognition routines (Khare et al., 2011). In the context of triangle listing in large graphs, tailored vertex orderings allow for precise minimization of cost functions $f^{++}(\pi)$ and $f^{+-}(\pi)$ that tightly predict and minimize algorithmic running time—though finding optimal orderings is NP-hard (Lécuyer et al., 2022).

In Cops and Robbers games, the existence of removable-vertex orderings is strictly equivalent to the existence of a Pursuer-winning strategy; explicit elimination orderings serve as strategy certificates and process guides (Bonato et al., 2017).

Moreover, ordering characterizations propagate closure properties under graph operators: for many classes defined by forbidden 3-vertex patterns, the class is closed under line-graph square, and orderings can be "lifted" algorithmically (Habib et al., 2017).

5. Multi-Partite and Geometric Generalizations

Recent research extends vertex-ordering characterizations to multipartite and geometric settings:

• Interval $r$-graphs and Circular-Arc $r$-graphs: Two types of ordering characterizations—generalized interval (fill-in) and matrix (consecutive-ones/blocks)—have been established for interval and circular-arc $r$-graphs, each with precisely catalogued forbidden patterns that increase in combinatorial complexity with $r$ (Paul et al., 16 Jan 2026, Paul et al., 22 Sep 2025).
• Circular-Arc Bigraphs and $H$-graphs: Notions such as total-circular and bi-circular orderings on cyclic orderings of multipartite vertex sets characterize circular-arc bigraphs and their generalizations (Paul et al., 13 Mar 2025).
• Geometric Orderings: The framework extends to candidate ranking via geometric "vantage point" orderings in $\mathbb{R}^d$, with rigorous enumeration of realizable orderings in terms of dimension and point set geometry (Alon et al., 2023).

These domains demonstrate that vertex-ordering properties are not restricted to purely combinatorial contexts but encode intersection, covering, and representability phenomena in geometric and topological structures as well.

6. Recognition Complexity, Algorithmic Implications, and Structural Insights

The existence of efficient (e.g., linear or polynomial time) algorithms for recognizing and producing class-certifying orderings is a central theme. For many cases with "local" constraints (e.g., only upper or lower bounds, or matrix contiguity), greedy or dynamic-programming approaches suffice. For classes with forbidden pattern characterizations of bounded size (e.g., interval bigraphs, cocomparability graphs), efficient both recognition and model construction algorithms exist (Habib et al., 2017, Paul et al., 13 Mar 2025).

However, simultaneous enforcement of various local and global constraints introduces NP-hardness or even APX-hardness; e.g., triangle-listing ordering cost minimization, degree-bounded orderings for arborescence induction (Lécuyer et al., 2022, Borsik et al., 5 Sep 2025).

Vertex orderings often reveal deep connections to forbidden subgraphs, clique decompositions, poset/cocomparability dimensions, convex geometry, and shellability/topological properties in higher dimensions (Goeckner et al., 2024, Dwary et al., 27 Apr 2025).

7. Connections, Unification, and Major Applications

Vertex-ordering characterizations unify and extend classical results, encompassing (but not limited to):

• Chordal, comparability, cocomparability, and AT-free graphs (via orderings such as perfect elimination, transitive, umbrella-free, or AT-free orderings),
• Claw-free and convex geometry-induced classes (with BFS-derived orderings that ensure structural monotonicity and domination properties) (Beisegel, 2018),
• Split comparability graphs, where an explicit vertex-labelling (ordering) yields dimension and representability bounds (Dwary et al., 27 Apr 2025),
• Algorithmic productivity: fast algorithms for triangle listing, induced matching computation, recognition, inference, and sampling in both static and dynamic settings (Lécuyer et al., 2022, Habib et al., 2017, Khare et al., 2011).

These ordering principles and their characterization theorems provide transparent, often minimal, certificates for class membership, guide the design of efficient algorithms, and serve as a template for analyzing new combinatorial classes.

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References:

(Lécuyer et al., 2022, Borsik et al., 5 Sep 2025, Dwary et al., 27 Apr 2025, Alon et al., 2023, Paul et al., 13 Mar 2025, Khare et al., 2011, Paul et al., 16 Jan 2026, Beisegel, 2018, Habib et al., 2017, Goeckner et al., 2024, Bonato et al., 2017, Paul et al., 22 Sep 2025, Takaoka, 2016)