Complete Split Graph: Theory & Applications

• Complete split graphs are split graphs whose vertices are divided into a clique and an independent set with every possible cross edge present.
• They play a key role in diverse fields such as spectral graph theory, polyhedral combinatorics, sandpile dynamics, and extremal graph problems.
• Their dual definitions offer insights into algorithmic efficiency, convex hull formulations, and algebraic invariants like Cohen–Macaulayness.

Searching arXiv for recent and foundational papers on complete split graphs and closely related split-graph literature. A complete split graph is, in several recent works, a split graph whose vertex set is partitioned into a clique and an independent set, with every possible edge between the two parts present; equivalently, $B_{p,q}:=K_p\nabla qK_1$, or, with parts $C$ and $S$, $E(G)=E(K_C)\cup \{(x,y):x\in C,\ y\in S\}$ (Wang et al., 17 Aug 2025, Fröberg, 2 Jun 2026). As a subclass of split graphs, it is simultaneously close to a complete graph and to a complete join with an independent set, and it appears in spectral graph theory, polyhedral combinatorics, commutative algebra, sandpile theory, graph pebbling, and extremal problems. The terminology is not completely uniform: in the classification of split graphs with four distinct eigenvalues, “the complete split graph” denotes the corona $K_c\circ K_1$, a different and narrower construction (Goldberg et al., 2014).

1. Terminology, models, and notational conventions

The join-based model is the dominant one in the literature considered here. If $C=\{x_1,\ldots,x_n\}$ is the complete part and $S=\{y_1,\ldots,y_m\}$ is the stable part, then a complete split graph is characterized by the property that every vertex in $S$ is adjacent to every vertex in $C$; equivalently, $n_i=|N_G(x_i)\cap S|=m$ for all $C$0 (Fröberg, 2 Jun 2026). In extremal graph theory this same object is denoted $C$1 and is also called a generalized book graph (Wang et al., 17 Aug 2025). In sandpile theory the notation $C$2 or $C$3 is used for the same clique-plus-independent-set join (Dukes, 2020, Derycke et al., 2024).

Notation	Definition	Context
$C$4	clique joined to $C$5 isolated vertices	forbidden-subgraph and spectral extremal problems
$C$6, $C$7	clique part plus independent part with all cross edges	Abelian sandpile model
$C$8	pendant vertex attached to each clique vertex	spectral paper using a narrower meaning of “complete split graph”

The corona-based usage is structurally distinct. In that setting, the graph has equal clique and stable set sizes, every clique vertex is connected to a unique pendant vertex from the independent set, all clique vertices have degree $C$9, all independent-set vertices have degree $S$0, and the graph is isomorphic to $S$1 (Goldberg et al., 2014). The coexistence of these conventions is a substantive feature of the literature rather than a notational accident.

2. Position inside split graph theory

A split graph is a graph whose vertex set can be partitioned into a clique and a stable set; such a partition is a KS-partition (Collins et al., 2017). Complete split graphs are a particular case in which the bipartite interface between the two parts is itself complete (Fröberg, 2 Jun 2026). Because they are split graphs, they lie in a class with several standard structural characterizations: split graphs are exactly the $S$2-free graphs, and they are exactly the chordal-$S$3-generalized split graphs (Brandstädt et al., 2017). The same $S$4-polar viewpoint appears in polarity theory, where graphs admitting a $S$5-polar partition are precisely split graphs (Mendoza et al., 2023).

The general split-graph framework also distinguishes balanced and unbalanced instances. A split graph is balanced if there exists a KS-partition with $S$6 and $S$7; it is unbalanced if every KS-partition fails to achieve both equalities simultaneously (Collins et al., 2017). Hammer and Simeone’s theorem, as summarized there, yields the trichotomy between balanced partitions, unbalanced $S$8-max partitions, and unbalanced $S$9-max partitions, with a swing vertex witnessing the unbalanced cases (Collins et al., 2017). This places complete split graphs inside the broader combinatorics of KS-partitions, clique number, and independence number, although the provided sources do not assign them uniformly to one side of the balanced/unbalanced dichotomy.

Split graphs are also closed under complementation, and by the Strong Perfect Graph Theorem they are perfect (Collins et al., 2017). A plausible implication is that complete split graphs inherit the algorithmic and structural tractability usually associated with perfect subclasses, but the concrete statements in the cited works concern particular problems rather than a single umbrella theorem.

3. Spectral viewpoints and competing meanings

The sharpest terminological divergence occurs in spectral graph theory. In the paper on split graphs with four distinct eigenvalues, the “complete split graph” is the corona $E(G)=E(K_C)\cup \{(x,y):x\in C,\ y\in S\}$0, not the join $E(G)=E(K_C)\cup \{(x,y):x\in C,\ y\in S\}$1 (Goldberg et al., 2014). That paper studies connected split graphs of diameter $E(G)=E(K_C)\cup \{(x,y):x\in C,\ y\in S\}$2 with exactly four distinct eigenvalues and proves that a connected bidegreed split graph of diameter $E(G)=E(K_C)\cup \{(x,y):x\in C,\ y\in S\}$3 has exactly four distinct eigenvalues if and only if it is either $E(G)=E(K_C)\cup \{(x,y):x\in C,\ y\in S\}$4 or a split graph $E(G)=E(K_C)\cup \{(x,y):x\in C,\ y\in S\}$5 arising from a $E(G)=E(K_C)\cup \{(x,y):x\in C,\ y\in S\}$6-design with $E(G)=E(K_C)\cup \{(x,y):x\in C,\ y\in S\}$7, $E(G)=E(K_C)\cup \{(x,y):x\in C,\ y\in S\}$8, $E(G)=E(K_C)\cup \{(x,y):x\in C,\ y\in S\}$9, and at least one pair of disjoint blocks (Goldberg et al., 2014). In this classification, the corona family is the case $K_c\circ K_1$0, with equal clique and stable-set sizes and pendant stable-set vertices.

The same paper records the standard block form of the adjacency matrix for a connected bidegreed split graph,

$K_c\circ K_1$1

and relates the spectrum to the eigenvalues of $K_c\circ K_1$2 (Goldberg et al., 2014). Within that framework, the corona-type complete split graphs form one infinite family of $K_c\circ K_1$3-extremal split graphs.

A different spectral role is played by the join-based complete split graph $K_c\circ K_1$4. In the spectral extremal problem for non-$K_c\circ K_1$5-partite graphs without complete split subgraphs, $K_c\circ K_1$6 is the forbidden graph, also called the generalized book graph (Wang et al., 17 Aug 2025). For $K_c\circ K_1$7, $K_c\circ K_1$8, and sufficiently large $K_c\circ K_1$9, the unique graph in $C=\{x_1,\ldots,x_n\}$0 is $C=\{x_1,\ldots,x_n\}$1, and $C=\{x_1,\ldots,x_n\}$2 (Wang et al., 17 Aug 2025). Thus complete split graphs appear both as extremal objects and as forbidden subgraphs in adjacency-spectral theory, but not always under a single fixed definition.

4. Convex hulls and extended formulations

For the join-based complete split graph, polyhedral combinatorics supplies an explicit description of the convex hull of the graph of the quadratic function

$C=\{x_1,\ldots,x_n\}$3

If $C=\{x_1,\ldots,x_n\}$4 is a complete split graph with clique $C=\{x_1,\ldots,x_n\}$5 of size $C=\{x_1,\ldots,x_n\}$6 and independent set $C=\{x_1,\ldots,x_n\}$7 of size $C=\{x_1,\ldots,x_n\}$8, then

$C=\{x_1,\ldots,x_n\}$9

and the target set is

$S=\{y_1,\ldots,y_m\}$0

The main theorem gives an extended formulation $S=\{y_1,\ldots,y_m\}$1, with projection $S=\{y_1,\ldots,y_m\}$2, where $S=\{y_1,\ldots,y_m\}$3 is defined by McCormick inequalities together with a family of clique inequalities (Harris et al., 2020).

Using the notation $S=\{y_1,\ldots,y_m\}$4, $S=\{y_1,\ldots,y_m\}$5, and $S=\{y_1,\ldots,y_m\}$6, the defining inequalities are

$S=\{y_1,\ldots,y_m\}$7

$S=\{y_1,\ldots,y_m\}$8

and

$S=\{y_1,\ldots,y_m\}$9

for $S$0, $S$1, and $S$2 (Harris et al., 2020). The paper emphasizes that these inequalities are sufficient for a tight formulation of the convex hull and contrasts this with even wheels, where McCormick plus triangle inequalities suffice (Harris et al., 2020).

The proof also provides a greedy, explicit construction of measurable sets $S$3 for a lifting argument via the Zuckerberg method, with $S$4 for $S$5 and a greedy overlap-minimizing construction for $S$6 (Harris et al., 2020). This makes the complete split graph one of the graph classes for which minimal extended formulations of this quadratic hull are given explicitly.

5. Edge rings, Betti numbers, and Cohen–Macaulayness

The algebraic theory of split graphs also singles out complete split graphs as a tractable extremal subclass. For a split graph with complete part $S$7, stable part $S$8, and $S$9, the Betti numbers of the edge ring $C$0 depend only on the multiset of the numbers $C$1, and the only nonzero Betti numbers are $C$2 and $C$3, $C$4 (Fröberg, 2 Jun 2026).

For a complete split graph, $C$5 for all $C$6, so the specialization becomes

$C$7

and the paper also records the equivalent Singh–Verma formula

$C$8

These formulas express the homological invariants entirely in terms of the partition sizes $C$9 (Fröberg, 2 Jun 2026).

The same work determines when the edge ring $n_i=|N_G(x_i)\cap S|=m$0 is Cohen–Macaulay: this holds if and only if no $n_i=|N_G(x_i)\cap S|=m$1 has a neighbor in $n_i=|N_G(x_i)\cap S|=m$2, or each $n_i=|N_G(x_i)\cap S|=m$3 has exactly one neighbor in $n_i=|N_G(x_i)\cap S|=m$4 (Fröberg, 2 Jun 2026). Since in a complete split graph each $n_i=|N_G(x_i)\cap S|=m$5 has $n_i=|N_G(x_i)\cap S|=m$6 neighbors in $n_i=|N_G(x_i)\cap S|=m$7, the edge ring is Cohen–Macaulay only if $n_i=|N_G(x_i)\cap S|=m$8 (Fröberg, 2 Jun 2026). This gives a clean contrast between combinatorial regularity and algebraic depth: the maximally joined split structure is typically not Cohen–Macaulay.

6. Sandpile dynamics, pebbling, and biclique partitions

The complete split graph $n_i=|N_G(x_i)\cap S|=m$9 supports a detailed Abelian sandpile theory. It consists of a clique part $C$00, an independent-set part $C$01, and all edges between the two parts (Dukes, 2020). The classification of recurrent states depends on the location of the sink. When the sink lies in the clique, weakly decreasing recurrent states are in bijection with Motzkin words of type $C$02, and the number of such decreasing recurrent configurations is

$C$03

When the sink lies in the independent set, the weakly decreasing recurrent states correspond to DH-Motzkin words, and

$C$04

The total number of spanning trees is

$C$05

obtained via a bijective Prüfer code argument (Dukes, 2020).

This program is extended by introducing two toppling conventions, CTI and ITC, on sorted recurrent configurations of $C$06, together with the statistics $C$07 and $C$08 (Derycke et al., 2024). Under a modification of the earlier bijection to Schröder paths, the bistatistic $C$09 maps to $C$10, and the $C$11-ITC polynomial becomes the $C$12-Schröder polynomial, implying symmetry in $C$13 and $C$14 (Derycke et al., 2024). The same paper characterizes sorted recurrent configurations by a new class of sawtooth polyominoes and proves a cyclic lemma yielding

$C$15

(Derycke et al., 2024).

Other graph invariants simplify sharply on complete split graphs. In pebbling theory, a complete split graph is a split graph where every vertex in the independent set is adjacent to every vertex in the clique, and its pebbling number is

$C$16

equivalently, complete split graphs are always Class $C$17 (Alcón et al., 2012). In biclique partition theory, for any split graph $C$18,

$C$19

and for a complete split graph with clique part $C$20 and independent set $C$21, the complement has $C$22 maximal cliques, so

$C$23

(Babu et al., 10 Jul 2025). In that construction, the complement consists of isolated vertices corresponding to $C$24 and a clique on $C$25.

7. Algorithmic and extremal roles

Complete split graphs also occur as boundary cases in algorithmic complexity. For the Steiner Tree problem, split graphs form a class with a sharp dichotomy: Steiner Tree is polynomial-time solvable on $C$26-free split graphs and NP-complete on $C$27-free split graphs (Illuri et al., 2015). Within this landscape, complete split graphs are singled out as a trivial case: if $C$28 is a terminal set in the independent part, then choosing any $C$29 connects all of $C$30, so a Steiner tree can be found in linear time (Illuri et al., 2015).

In forbidden-subgraph extremal theory, the join-based complete split graph reappears as the obstruction $C$31, also called the generalized book graph (Wang et al., 17 Aug 2025). For sufficiently large $C$32, the unique spectral extremal $C$33-vertex non-$C$34-partite $C$35-free graph is $C$36, obtained from $C$37 by deleting one edge $C$38 and adding a new vertex adjacent to all of $C$39 and to $C$40; moreover,

$C$41

with equality if and only if $C$42, and $C$43 (Wang et al., 17 Aug 2025). This places complete split graphs among the small family of concrete forbidden graphs for which spectral and edge extremal problems are both resolved.

More broadly, split graphs are the base level of several hierarchies. They are chordal-$C$44-generalized split graphs (Brandstädt et al., 2017) and $C$45-polar graphs (Mendoza et al., 2023). A plausible implication is that complete split graphs serve as canonical test objects in these generalized frameworks: they realize the pure clique–stable-set decomposition with maximal interaction across the partition, while retaining the finite obstruction and efficient-recognition phenomena inherited from split-graph theory.