Generalized Circular Chord Graphs

• Generalized circular chord graphs are graph models that use cyclic orders on circles or rings to encode adjacency via chord intersections and modular rules.
• They employ matrix formalisms, forbidden subgraph characterizations, and transfer-matrix techniques to reveal spectral properties and algorithmic structures.
• Applications include network design, cyclic scheduling, Gaussian graph realizability, and topological extensions in graph drawing and knot theory.

Generalized circular chord graphs form a family of graph models in which cyclic order on a circle—or on one or several rings—governs adjacency, intersection, coloring, or metric structure. The umbrella includes classical circle graphs, where vertices correspond to chords and adjacency records chord intersection; split and circular-arc restrictions of those models; fixed-offset chorded cycles such as $\mathcal C_n^{(k)}$; ring and multi-ring families built from modular chord rules; and surface-based interlace graphs arising from chord diagrams of immersed loops. Across these formulations, the common theme is that a circular ordering yields rigid matrix patterns, parity constraints, and often strong algorithmic or spectral structure (Bonomo-Braberman et al., 2020, Lopez-Bonilla et al., 6 Sep 2025, Reyes et al., 2024, Simon, 2023).

1. Core definitions and scope

Classically, a graph $G=(V,E)$ is a circle graph if there exists a set of chords $\mathcal C$ on a circle and a bijection $\phi:V\to\mathcal C$ such that

$uv\in E \iff \phi(u)\cap \phi(v)\neq\emptyset.$

In probabilistic work, a chord diagram of size $n$ is a pairing of $2n$ labeled points on a circle, and its intersection graph is defined by chord crossings. In fixed-offset cyclic work, generalized circular chord graphs $\mathcal C_n^{(k)}$ are graphs on $\mathbb Z_n$ with cycle edges, chords of length $k$, and, when $G=(V,E)$0 is even, diameters. In network-oriented work, chordal ring, chordal multi-ring, and mixed variants place vertices on one or several cycles and add parity-dependent modular chords. In topological work, a chordiagraph is the interlace graph of a chord diagram associated with a filoop on a surface (Bonomo-Braberman et al., 2020, Acan, 2015, Lopez-Bonilla et al., 6 Sep 2025, Reyes et al., 2024, Simon, 2023).

These definitions encode different emphases. In circle graphs and random chord-diagram models, the central relation is geometric intersection of chords. In $G=(V,E)$1, chordal rings, and maximal outerplanar triangulations, the central datum is cyclic distance along the boundary or along a modular cycle. In filoop and Gauss-diagram settings, the same chord combinatorics is enriched by orientation, framing, or genus. This suggests that “generalized circular chord graphs” is best understood as a research umbrella rather than a single canonical graph class.

A recurring neighboring notion is the split graph $G=(V,E)$2, where $G=(V,E)$3 induces a clique and $G=(V,E)$4 an independent set. Another is the circular-arc graph, defined as the intersection graph of arcs on a circle. These subclasses are important because the circular order then interacts with clique–stable decompositions or interval representations in especially rigid ways.

2. Structural characterizations in restricted subclasses

Within split graphs, circle-graph recognition admits a forbidden induced subgraph characterization. If $G=(V,E)$5 denotes the family consisting of tent $G=(V,E)$6, odd $G=(V,E)$7-suns with center, even $G=(V,E)$8-suns, the matrix-defined families $G=(V,E)$9, $\mathcal C$0, $\mathcal C$1, $\mathcal C$2, and the families $\mathcal C$3, $\mathcal C$4, $\mathcal C$5, then a split graph is a circle graph if and only if it contains none of the graphs in $\mathcal C$6 as an induced subgraph; every member of $\mathcal C$7 is minimally non-circle. The structural analysis proceeds by fixing induced tent, 4-tent, or co-4-tent configurations and partitioning $\mathcal C$8 and $\mathcal C$9 so that vertices of $\phi:V\to\mathcal C$0 acquire circularly monotone neighborhoods in ordered clique blocks (Bonomo-Braberman et al., 2020).

A distinct but closely related line concerns split graphs inside circular-arc graphs. McConnell’s transformation associates to a chordal graph $\phi:V\to\mathcal C$1 and clique $\phi:V\to\mathcal C$2 a graph $\phi:V\to\mathcal C$3. In the split case, $\phi:V\to\mathcal C$4 is circular-arc if and only if there exists $\phi:V\to\mathcal C$5 such that $\phi:V\to\mathcal C$6 is an interval graph with an interval model in which no interval for a vertex in $\phi:V\to\mathcal C$7 contains an interval for a vertex in $\phi:V\to\mathcal C$8. For Helly circular-arc split graphs the criterion simplifies to: $\phi:V\to\mathcal C$9 is Helly circular-arc if and only if $uv\in E \iff \phi(u)\cap \phi(v)\neq\emptyset.$0 is interval. This yields explicit minimal split obstructions, including the long-claw-derived and whipping-top-derived graphs, the families $uv\in E \iff \phi(u)\cap \phi(v)\neq\emptyset.$1 and $uv\in E \iff \phi(u)\cap \phi(v)\neq\emptyset.$2, and the graphs $uv\in E \iff \phi(u)\cap \phi(v)\neq\emptyset.$3 obtained by adding a vertex complete to the clique side of $uv\in E \iff \phi(u)\cap \phi(v)\neq\emptyset.$4 (Cao et al., 2024).

These characterizations are sharply subclass-specific. A full forbidden induced subgraph characterization of all circle graphs is still unknown. The split obstruction list also does not extend verbatim even to all chordal graphs: the graph $uv\in E \iff \phi(u)\cap \phi(v)\neq\emptyset.$5 is chordal but neither split nor circle, so additional obstructions are needed outside the split setting.

3. Matrix formalisms and recognition methods

A major methodological theme is the translation of circular geometry into matrix structure. For a split graph $uv\in E \iff \phi(u)\cap \phi(v)\neq\emptyset.$6, the incidence matrix $uv\in E \iff \phi(u)\cap \phi(v)\neq\emptyset.$7 records adjacency between the stable and clique sides. A matrix is nested if its columns can be permuted so that every row has the consecutive-ones property and any two rows are either disjoint or nested; equivalently, it contains no 0-gem subconfiguration

$uv\in E \iff \phi(u)\cap \phi(v)\neq\emptyset.$8

Circle-graph constraints inside split graphs require the richer notion of an enriched 2-nested matrix, with LR-labeled rows, block bi-coloring, LR-orderings, and exclusion of several local subconfiguration families together with monochromatic or badly colored gem patterns. These conditions encode precisely when chord endpoints can be placed on a circle so that crossings reproduce adjacency (Bonomo-Braberman et al., 2020).

On the algorithmic side, classical circle-graph recognition already has Naji’s algebraic characterization, yielding an $uv\in E \iff \phi(u)\cap \phi(v)\neq\emptyset.$9 algorithm, and an almost-linear-time recognition algorithm due to Geelen, Paul, Tedder, and Corneil. In the split subclass, the forbidden-family and 2-nestedness machinery imply polynomial-time recognition. For split circular-arc graphs, the interval-pattern viewpoint goes further: a linear-time certifying recognition algorithm either outputs a circular-arc model or an induced minimal obstruction, using the transformation $n$0, interval recognition, and explicit obstruction lifting (Cao et al., 2024).

Realizability of Gauss diagrams leads to a parallel algebraic formalism over $n$1. If $n$2 is the adjacency matrix of the circle graph of a Gauss diagram, realizability is equivalent to the existence of a diagonal matrix $n$3 such that $n$4 is idempotent, or, equivalently, to solvability of

$n$5

In graph-theoretic terms, the circle graph must be Eulerian and its edge weights $n$6 must satisfy the weighted cycle condition $n$7 for every cycle. The resulting recognition procedure checks parity on non-adjacent pairs and reduces the remaining constraints to bipartiteness of an auxiliary graph (Khan et al., 2021).

4. Random, extremal, and enumerative phenomena

Uniform random chord diagrams provide the canonical probabilistic model. A chord diagram of size $n$8 is a perfect matching on $n$9, chosen uniformly among

$2n$0

possibilities, and its intersection graph is $2n$1. If $2n$2 for the chord containing endpoint $2n$3, then

$2n$4

For $2n$5, the $2n$6-core is, with high probability, exactly the subdiagram $2n$7 of chords of length at least $2n$8. Fixed-length counts satisfy $2n$9, the number of trivial strong components in a random orientation converges to $\mathcal C_n^{(k)}$0, and, with high probability, the evolving diagram becomes monolithic and then stays monolithic (Acan, 2015).

A different extremal theme studies cyclic distance rather than intersection. In a maximal outerplanar graph drawn on an outer cycle $\mathcal C_n^{(k)}$1, each non-cycle edge is a chord of combinatorial length

$\mathcal C_n^{(k)}$2

and the total chord length is

$\mathcal C_n^{(k)}$3

For $\mathcal C_n^{(k)}$4, the minimum is

$\mathcal C_n^{(k)}$5

achieved by the greedy graphs $\mathcal C_n^{(k)}$6. The maximum is

$\mathcal C_n^{(k)}$7

achieved by the shell graphs $\mathcal C_n^{(k)}$8. A maximal outerplanar graph has maximal total chord length if and only if it has exactly two ears, and every integer between the minimum and maximum values occurs as the total chord length of some maximal outerplanar graph of order $\mathcal C_n^{(k)}$9 (Broadus et al., 2024).

Enumeration results expose the size of the realizable subclass. The number of realizable circle graphs on $\mathbb Z_n$0 crossings is

$\mathbb Z_n$1

for $\mathbb Z_n$2, and these graphs coincide with alternating prime knots modulo mutation. The meander subclass, obtained from open meanders by closing with a chord that intersects all others, has idempotent adjacency matrix $\mathbb Z_n$3; the corresponding counts of meander graphs are

$\mathbb Z_n$4

at $\mathbb Z_n$5 (Khan et al., 2021).

5. Fixed-offset, circulant, and ring-based families

One explicit use of the term generalized circular chord graphs defines $\mathbb Z_n$6 on vertex set $\mathbb Z_n$7 by adjoining to the cycle $\mathbb Z_n$8 all edges $\mathbb Z_n$9 and, when $k$0 is even, all diameters $k$1. For fixed $k$2 and number of colors $k$3, the chromatic counts satisfy

$k$4

for a finite transfer matrix $k$5. In the case $k$6 and odd $k$7, the exact formula is

$k$8

where $k$9 is the Lucas sequence and $G=(V,E)$00. Consequently,

$G=(V,E)$01

with $G=(V,E)$02. For even $G=(V,E)$03, diameter constraints are handled by a paired-window transfer matrix $G=(V,E)$04 of size at most $G=(V,E)$05, and the resulting counts exhibit modular patterns across residue classes without universal vanishing rules (Lopez-Bonilla et al., 6 Sep 2025).

Network-oriented generalizations replace one circle by one or several rings with modular chord rules. The undirected chordal ring $G=(V,E)$06 and chordal multi-ring $G=(V,E)$07 are 3-regular and bipartite, while the mixed families $G=(V,E)$08 and $G=(V,E)$09 replace ring edges by directed arcs and keep one undirected chord per vertex. Plane tessellations of the infinite patterns yield order-diameter bounds. For $G=(V,E)$10, the maximum number of vertices of diameter $G=(V,E)$11 is $G=(V,E)$12 for odd $G=(V,E)$13 and $G=(V,E)$14 for even $G=(V,E)$15; for even $G=(V,E)$16, $G=(V,E)$17 attains $G=(V,E)$18. In the mixed case, the corresponding upper bound is $G=(V,E)$19 for odd $G=(V,E)$20 and $G=(V,E)$21 for even $G=(V,E)$22. These families are also lifts over Abelian groups, which leads to closed formulas for the adjacency spectrum, with undirected spectra parameterized by Fourier modes and mixed spectra generally complex (Reyes et al., 2024).

These fixed-offset and multi-ring models supply concrete applications. The $G=(V,E)$23 coloring formulas are presented as feasibility engines for cyclic scheduling problems such as airline gate assignment, wireless sensor networks, and multiprocessor task coordination. Chordal rings and multi-rings are used as interconnection-network topologies in which all nodes lie on one or several cycles. A plausible implication is that “generalized circular chord graphs” has become a meeting point between transfer-matrix combinatorics, network design, and circulant spectral theory.

6. Topological and minor-theoretic extensions

Generalization to surfaces replaces planar chord diagrams by framed chord diagrams coming from filoops, that is, generic immersions of a circle in a closed oriented surface whose complement is a disjoint union of discs. The interlace graph of such a chord diagram is called a chordiagraph. For graphs of even degrees, the quantity $G=(V,E)$24 gives the minimal genus among filoops having interlace graph $G=(V,E)$25; in the bicolourable case,

$G=(V,E)$26

where $G=(V,E)$27 is the Rosenstiehl form. The case $G=(V,E)$28 characterizes Gaussian graphs, precisely those interlace graphs that admit genus-0 framings and hence spheriloops; if $G=(V,E)$29 is connected and Gaussian, every chord diagram with interlace graph $G=(V,E)$30 has exactly two genus-0 framings (Simon, 2023).

The same surface framework supports canonical factorization. Spheric sums correspond to disjoint unions of interlace graphs, while essential plumbings correspond to splits in the Cunningham decomposition of a connected graph. The genus is additive under both operations, and Gaussian graphs can be generated by an unambiguous context-sensitive grammar built from CL2- and CL12-weighted graph-labelled trees. This places generalized circular chord graphs inside an operadic picture in which local complementation, split decomposition, and genus interact systematically.

A complementary direction studies circle graphs and more general circular drawings through treewidth. Large-treewidth circle graphs contain induced subgraphs formed by $G=(V,E)$31 vertex-disjoint cycles $G=(V,E)$32 such that every vertex of $G=(V,E)$33 has at least two neighbours in $G=(V,E)$34 for $G=(V,E)$35, and every vertex has at most four neighbours in any fixed $G=(V,E)$36. Consequently, treewidth and Hadwiger number are linearly tied on the class of circle graphs, while Hajós number is quadratically tied to both. For an arbitrary graph $G=(V,E)$37 with a circular drawing $G=(V,E)$38, if the crossing graph $G=(V,E)$39 has no $G=(V,E)$40-minor then $G=(V,E)$41 and $G=(V,E)$42 has no $G=(V,E)$43-topological minor; by contrast, there are graphs with arbitrarily large Hadwiger number that admit circular drawings whose crossing graphs are 2-degenerate (Hickingbotham et al., 2022).

Several problems remain open across these strands. A full forbidden induced subgraph characterization of all circle graphs is still unknown, and even the characterization of circle graphs within chordal graphs remains open. In the fixed-offset family $G=(V,E)$44, the spectrum of the paired-window matrix $G=(V,E)$45 for even $G=(V,E)$46 has not been fully analyzed. For cyclic-length invariants, natural extensions include weighted boundary metrics and non-maximal outerplanar graphs (Bonomo-Braberman et al., 2020, Lopez-Bonilla et al., 6 Sep 2025, Broadus et al., 2024).