Interval r-Graphs: Theory & Recognition

• Interval r-Graphs are r-partite graphs with a real interval representation, where vertices in different parts are adjacent if and only if their intervals intersect.
• They are characterized by precise vertex-ordering criteria and forbidden subordering patterns that facilitate structural analysis and algorithmic recognition.
• Research also examines interval spectra in r-regular graphs, establishing bounds and revealing that the induced subgraphs of interval-spectrum vertices form path-like forests.

An interval $r$-graph is a type of $r$-partite graph admitting a real interval representation such that adjacency occurs exactly between vertices in different parts whose assigned intervals intersect. The class of interval $r$-graphs generalizes the classical interval graphs ($r=1$) and interval bigraphs ($r=2$), and their study links graph theory, combinatorial optimization, and computational recognition problems. Recent work provides rigorous vertex-ordering characterizations, adjacency-matrix criteria, forbidden pattern descriptions, and implications for structure and algorithmic recognition (Paul et al., 16 Jan 2026). Separately, notions of "interval spectrum" for edge-labeled regular graphs connect through distinct, though structurally related, combinatorial invariants (Davtyan et al., 2013).

1. Formal Definitions

Let $r\ge2$. An $r$-partite graph $G=(V,E)$ is one whose vertex set can be partitioned into independent sets $X_1,\dots,X_r$, i.e., $V = X_1\cup\cdots\cup X_r$ with $X_i\cap X_j=\emptyset$ for $i\neq j$, and $E$ contains no edges within any $X_i$.

$G$ is an interval $r$-graph if there exists a family $\{I_v\subset\mathbb{R} : v\in V\}$ of closed intervals such that for any $u\in X_\alpha$, $v\in X_\beta$, $\alpha\ne\beta$, we have $\{u,v\}\in E$ if and only if $I_u\cap I_v\ne\emptyset$. No adjacency is permitted within parts. This model ensures that edges occur precisely when the corresponding intervals for different parts overlap (Paul et al., 16 Jan 2026).

Separately, for a simple (not necessarily multipartite) $r$-regular graph $G$ and an injective edge labeling $\varphi:E(G)\to\{1,2,\dots,|E(G)|\}$, define the spectrum $S_G(x,\varphi)$ of a vertex $x$ as the set of incident edge labels. Vertices for which this spectrum is an interval of integers form the set $V_{\rm int}(G,\varphi)$; this set interacts with combinatorial properties of $G$ and leads to sharp vertex-count inequalities (Davtyan et al., 2013).

2. Vertex Ordering Characterizations

Central to the structure theory are two vertex-ordering-based criteria that are equivalent to being an interval $r$-graph (Paul et al., 16 Jan 2026):

• Generalized interval ordering: $G$ admits a linear ordering $v_1,\dots,v_n$ such that for each edge $v_iv_j\in E$ ($i>j$), $v_i\in X_\alpha$, $v_j\in X_\beta$, $\alpha\neq\beta$, and every $v_\ell$ with $j<\ell<i$ not in $X_\beta$ must also be adjacent to $v_j$. Formally,

$$v_i v_j\in E,\,i>j \implies \forall\,j<\ell<i\ \text{with}\ v_\ell\notin X_\beta:\ v_\ell v_j\in E.$$

Existence of such an ordering is equivalent to $G$'s being an interval $r$-graph.

• $r$-interval ordering (almost consecutive ones): In the adjacency matrix ordered by $v_1,\dots,v_n$, for each row $i$ (vertex $v_i$), the block of 1’s (entries for edges to higher-index vertices in different parts) must be contiguous, except that zeros are permitted where corresponding to vertices in the same part. A similar property must hold for columns. The union of all such row and column blocks must cover precisely the 1's of the adjacency matrix.

These characterizations not only provide structural insight but also facilitate the design of recognition algorithms (Paul et al., 16 Jan 2026).

3. Forbidden Patterns

For $r\ge3$, interval $r$-graphs admit a forbidden subordering description in terms of three-vertex patterns. Fix an ordering $v_1<\cdots<v_n$. Forbidden configurations comprise:

1. Three vertices in distinct parts at positions $i<j<k$, with only $v_iv_k$ an edge.
2. First and last vertices in the same part, middle in a different part, with only the first–last edge.
3. Three vertices in distinct parts, with only the last two forming an edge.

Explicitly, these local patterns forbid certain isolated adjacencies in triple sequences, which cannot be realized by the interval representation; violation of this criterion prevents interval $r$-graph structure. This characterization leads to an $O(n^3)$ recognition algorithm by exhaustive checking of all ordered triples, though for small $r$ and sparse graphs, practical performance may be better (Paul et al., 16 Jan 2026).

4. Edge Labeling, Interval Spectrum, and Bounds in Regular Graphs

In contrast to adjacency models, examining spectra of edge labelings in regular graphs leads to the following results (Davtyan et al., 2013):

• For an $r$-regular graph $G$ ($r\ge2$) and injective edge labeling, the subset $V_{\rm int}(G,\varphi)$ (vertices with interval spectrum) induces a subgraph $H$. $H$ must be a union of disjoint paths—a forest (no cycles).
• The cardinality of $V_{\rm int}(G,\varphi)$ satisfies

$$|V_{\rm int}(G,\varphi)| \le \frac{r|V(G)| - 2\eta(H)}{2(r-1)},$$

where $\eta(H)$ is the number of components of $H$.

• In cubic graphs $(r=3)$:

$$|V_{\rm int}(G,\varphi)| \le \left\lfloor \frac{3|V(G)|-2\eta(H)}{4} \right\rfloor.$$

• For $r=2$, $G$ is a union of cycles and $|V_{\rm int}| \le |V(G)|-1$.
• Corollary: For $r$-regular $G$, $$|V_{\rm int}(G,\varphi)| \le \frac{2(r-1)}{r}\,|V(G)|$$.

These results reveal a sharp limitation: not all regular graphs admit global interval spectrum; the induced structure is highly constrained—always acyclic and path-structured (Davtyan et al., 2013).

5. Examples, Non-examples, and Structural Consequences

Interval $r$-graphs encompass all interval bigraphs ($r=2$), with classical consecutive-ones properties. For $r=3$ and higher, the order-forbidden substructures become decisive.

Example (interval 3-graph): Parts $X_1=\{v_1, v_6, v_{10}\}$, $X_2=\{v_2, v_8, v_9\}$, $X_3=\{v_3, v_4, v_5, v_7\}$, ordered $v_1,\dots,v_{10}$, with valid interval assignments and absence of all three forbidden patterns; adjacency and interval representations cohere (Paul et al., 16 Jan 2026).

Non-example: Three vertices $a, b, c$ in distinct parts, edge only $a$–$c$, ordered $(a,b,c)$—pattern (i) occurs; this cannot be modeled as an interval 3-graph.

For regular graphs and interval spectrum, only path-like induced subgraphs can arise; cycles of length $\geq3$ are precluded in $H$. The maximum number of interval-spectra vertices is strictly less than $V(G)$ unless specific structural conditions are met (Davtyan et al., 2013).

6. Algorithmic Aspects and Recognition

The characterization results yield straightforward recognition algorithms:

• Interval ordering search: Attempt to find vertex orderings fulfilling generalized or $r$-interval order constraints. Brute-force search requires $O(n^2)$ time, improved in practice with PQ-trees or greedy techniques, especially for small $r$ (Paul et al., 16 Jan 2026).
• Forbidden patterns: Checking all ordered triples for forbidden suborderings yields an $O(n^3)$ test, with certifying counterexamples.
• Certifying recognition: Any violation (hole in fill, forbidden triple) can serve as a certificate for non-membership in the class, paving the way for linear-time certifying algorithms for bounded $r$.

For interval spectrum questions in regular graphs, the existence or infeasibility of certain edge labelings may be determined by combinatorial inspection of the induced path substructure (Davtyan et al., 2013).

7. Open Questions and Connections

Several open questions have been articulated:

• Which $r$-regular graphs admit an edge labeling with all vertices interval spectrum?
• For fixed $r$, what is the maximum ratio $|V_{\rm int}(G,\varphi)|/|V(G)|$ achievable in an infinite family of $r$-regular graphs?
• Do the ordering-based structural characterizations for interval $r$-graphs extend to more general models, such as circular-arc representations or relaxations permitting gaps in interval spectra?

These directions connect interval $r$-graphs to broader themes in characterizing intersection graphs, forbidden order patterns, and combinatorial structure, linking structural theory with recognition complexity and matrix representation techniques (Paul et al., 16 Jan 2026, Davtyan et al., 2013).