Splitting Graphs in Algebra and Combinatorics

• Splitting graphs are finite graphs derived from a base graph via a surjection that preserves the edge set, serving as a bridge between combinatorial structure and algebraic invariants.
• They are used to compare edge ideals by analyzing invariants such as projective dimension, regularity, and Betti numbers under controlled vertex-splitting operations.
• The construction preserves some graph properties (like bipartiteness and forest structure) while potentially altering homological properties like depth, offering nuanced insights into graph theory and commutative algebra.

A splitting graph, in the sense introduced by Herzog, Moradi, and Rahimbeigi, is a finite simple graph $\widetilde G$ attached to a finite simple graph $G$ by a surjection on vertices that preserves the edge set of $G$ exactly at the level of images. Its purpose is algebraic as much as combinatorial: the construction is designed to compare the edge ideal

$$I(G)=\bigl(x_i x_j:\{i,j\}\in E(G)\bigr)\subset S=K[x_i:i\in V(G)]$$

with the edge ideal $$I(\widetilde G)\subset S'=K[y_v:v\in V(\widetilde G)]$$, and thereby to study how homological invariants behave under controlled vertex-splitting operations (Herzog et al., 2019). In the broader literature, however, the term “splitting graph” is not uniform: it also denotes a classical graph transformation $S(G)$, various $m$-splitting constructions, and, in geometric group theory, the free-splitting graph.

1. Formal definition and basic mechanism

Let $G$ be a finite simple graph. A graph $\widetilde G$ is called a splitting graph of $G$ if there exists a surjection

$G$0

such that for every edge $G$1, the image $G$2 is an edge of $G$3, and the induced map

$G$4

is bijective. The map $G$5 is called a splitting map. Intuitively, $G$6 “blows up” some vertices of $G$7 into several copies and reattaches edges so that one still sees precisely the same edge-set on the level of $G$8 (Herzog et al., 2019).

The algebraic setting is built from the edge ideals of the two graphs. If $G$9 has vertex set $G$0, then

$G$1

while for $G$2 one writes

$G$3

The central problem is then to compare invariants of $G$4 and $G$5. In this formulation, splitting graphs form a combinatorial device for transporting information between graphs and monomial ideals rather than merely a graph-editing operation.

2. Projective dimension, regularity, and conjectured monotonicity

The basic invariants considered in this setting are

$G$6

The guiding expectation is that passing from $G$7 to a splitting graph $G$8 should not improve these invariants. More precisely, one hopes always to have

$G$9

$$I(G)=\bigl(x_i x_j:\{i,j\}\in E(G)\bigr)\subset S=K[x_i:i\in V(G)]$$0

and

$$I(G)=\bigl(x_i x_j:\{i,j\}\in E(G)\bigr)\subset S=K[x_i:i\in V(G)]$$1

In full generality these statements remain conjectural (Herzog et al., 2019).

This monotonicity program places splitting graphs in a homological framework that is close to other comparison principles for edge ideals, but it is more delicate because the construction changes the ambient polynomial ring and the vertex set while preserving the edge set only through the splitting map. The conjectural inequalities therefore concern not only combinatorial complexity but also the way free resolutions respond to controlled duplication of vertices.

3. Special splittings and the inductive mechanism

A workable sufficient hypothesis is the notion of a special splitting. A splitting map $$I(G)=\bigl(x_i x_j:\{i,j\}\in E(G)\bigr)\subset S=K[x_i:i\in V(G)]$$2 is called special if one of the following holds:

1. Whenever $$I(G)=\bigl(x_i x_j:\{i,j\}\in E(G)\bigr)\subset S=K[x_i:i\in V(G)]$$3 satisfy $$I(G)=\bigl(x_i x_j:\{i,j\}\in E(G)\bigr)\subset S=K[x_i:i\in V(G)]$$4, every neighbor of $$I(G)=\bigl(x_i x_j:\{i,j\}\in E(G)\bigr)\subset S=K[x_i:i\in V(G)]$$5 in $$I(G)=\bigl(x_i x_j:\{i,j\}\in E(G)\bigr)\subset S=K[x_i:i\in V(G)]$$6 is adjacent to every neighbor of $$I(G)=\bigl(x_i x_j:\{i,j\}\in E(G)\bigr)\subset S=K[x_i:i\in V(G)]$$7.
2. Whenever $$I(G)=\bigl(x_i x_j:\{i,j\}\in E(G)\bigr)\subset S=K[x_i:i\in V(G)]$$8 lie in the same fiber $$I(G)=\bigl(x_i x_j:\{i,j\}\in E(G)\bigr)\subset S=K[x_i:i\in V(G)]$$9, they lie in different connected components of $$I(\widetilde G)\subset S'=K[y_v:v\in V(\widetilde G)]$$0.

Under this hypothesis, Herzog–Moradi–Rahimbeigi prove

$$I(\widetilde G)\subset S'=K[y_v:v\in V(\widetilde G)]$$1

The proof reduces a special splitting step-by-step to the merging of two vertices $$I(\widetilde G)\subset S'=K[y_v:v\in V(\widetilde G)]$$2 with the same image, and the main algebraic device is the short exact sequence

$$I(\widetilde G)\subset S'=K[y_v:v\in V(\widetilde G)]$$3

where $$I(\widetilde G)\subset S'=K[y_v:v\in V(\widetilde G)]$$4. In the first special case one shows combinatorially that $$I(\widetilde G)\subset S'=K[y_v:v\in V(\widetilde G)]$$5 under the special adjacency condition, while in the second case one uses decomposition into connected components and adds projective dimensions and regularity over disjoint sets of variables (Herzog et al., 2019).

A corollary applies the regularity inequality to several graph classes already known to satisfy $$I(\widetilde G)\subset S'=K[y_v:v\in V(\widetilde G)]$$6, where $$I(\widetilde G)\subset S'=K[y_v:v\in V(\widetilde G)]$$7 is the induced-matching number. The listed classes are sequentially Cohen–Macaulay, chordal, weakly chordal, sequentially CM bipartite, unmixed bipartite, very well-covered, and $$I(\widetilde G)\subset S'=K[y_v:v\in V(\widetilde G)]$$8-free vertex-decomposable graphs. In these cases,

$$I(\widetilde G)\subset S'=K[y_v:v\in V(\widetilde G)]$$9

and hence $S(G)$0.

4. Betti numbers, dimension, depth, and preserved graph classes

The behavior of Betti numbers is only partly understood. If condition (2) in the definition of special splitting holds, so that each fiber lies in distinct connected components, then one has

$S(G)$1

This is the case in which Betti-number monotonicity is proved rather than conjectured (Herzog et al., 2019).

For arbitrary splitting graphs, one always has the dimension inequality

$S(G)$2

If, in addition, $S(G)$3 is a path or an even cycle, then

$S(G)$4

However, depth inequalities fail in general. A counterexample in the same paper shows that splitting may lower depth by at least $S(G)$5, so Cohen–Macaulayness need not be preserved. Accordingly, even when $S(G)$6 is Cohen–Macaulay of one of the special types above, the monotonicity of projective dimension and regularity only says that $S(G)$7 has at least as bad projective dimension or regularity; it does not recover Cohen–Macaulayness.

Some purely graph-theoretic properties are preserved. If $S(G)$8 is bipartite, then any splitting graph $S(G)$9 is again bipartite. If $m$0 is a forest, then any splitting graph $m$1 is again a forest. These preservation results sharply contrast with the failure of depth monotonicity, and they show that the combinatorial effect of splitting can be mild even when the homological effect is not (Herzog et al., 2019).

5. The classical splitting graph $m$2

An older graph-theoretic construction uses the notation $m$3. If $m$4 is a finite simple graph of order $m$5, the splitting graph $m$6 is obtained by introducing a new vertex $m$7 for each $m$8, and joining $m$9 to $G$0 if and only if $G$1. Equivalently,

$G$2

where $G$3 and

$G$4

Here $G$5 and $G$6 (Castro et al., 24 Feb 2026).

In Sampathkumar–Walikar’s 1980 treatment, it was asserted that for any $G$7 of order $G$8,

$G$9

where $\widetilde G$0 is the vertex-cover number and $\widetilde G$1 the independence number. Castro, Leaños, and Rosario show that these equalities do not hold in general, provide counterexamples, and replace them with exact formulas. Defining

$\widetilde G$2

they prove that if $\widetilde G$3 is a connected simple graph of order $\widetilde G$4, then

$\widetilde G$5

Equivalently, the old formulas hold precisely when $\widetilde G$6, that is, precisely when $\widetilde G$7 for every independent set $\widetilde G$8 (Castro et al., 24 Feb 2026).

The corrected formulas imply the sharp range

$\widetilde G$9

and every integer in that interval is attained by a suitable graph of order $G$0. This line of work shows that the classical splitting graph is governed by neighborhood expansion of independent sets rather than by a universal equality at $G$1.

6. Terminological scope and adjacent research directions

The literature uses closely related names for several distinct constructions.

Term	Definition	Representative result
Splitting graph of $G$2	A graph $G$3 with a surjection $G$4 whose induced edge map is bijective	Comparison of $G$5, $G$6, Betti numbers, depth, and dimension (Herzog et al., 2019)
Classical splitting graph $G$7	Add one vertex $G$8 for each $G$9, with $G$00 adjacent to exactly the neighbors of $G$01 in $G$02	$G$03, $G$04 (Castro et al., 24 Feb 2026)
$G$05-splitting graph $G$06	Add $G$07 new copies $G$08 of each vertex-neighborhood pattern	$G$09 (Dudhat et al., 27 Mar 2026)
Free-splitting graph $G$10	Vertices are conjugacy classes of one-edge free splittings of $G$11	$G$12 is Gromov hyperbolic and $G$13 (Hamenstaedt, 2012)

A separate but nearby notion is the split graph: a graph whose vertex set can be partitioned into a clique and an independent set. This is not a splitting graph. Split graphs admit the Földes–Hammer forbidden-subgraph characterization by exclusion of induced $G$14, $G$15, and $G$16, and they further divide into balanced and unbalanced classes (Cheng et al., 2015).

There is also an extensive algorithmic literature on vertex splitting as a graph-modification operation. For transforming graphs into interval graphs, the decision problem “VC-Interval-Split” asks whether a graph can be turned into an interval graph using at most $G$17 vertex splits; this problem is NP-hard even on planar subcubic bipartite graphs, while splitting into a disjoint union of paths and splitting triangle-free graphs into unit interval graphs are polynomial-time solvable (Abu-Khzam et al., 4 Feb 2026). For plane graphs, the outerplane splitting number $G$18 is equivalent to the existence of a connected face cover of size $G$19, and also equivalent to a minimum feedback vertex set of size $G$20 in the dual; the associated decision problem is NP-complete for plane biconnected graphs, but maximal planar graphs admit a polynomial-time algorithm (Gronemann et al., 2023). For planarization by vertex splitting in abstract graphs and fixed drawings, the splitting number problem is NP-complete, Embedded Vertex Deletion and Split Set Re-Embedding are NP-complete, the abstract problem is non-uniformly fixed-parameter tractable in the number of splits, and Split Set Re-Embedding can be solved in time $G$21 (Nöllenburg et al., 2022).

Taken together, these lines of work show that “splitting graph” is not a single universally standardized term. In commutative algebra it denotes a controlled lift of a graph preserving the edge set through a splitting map; in classical graph theory it denotes the construction $G$22; in spectral graph theory it includes $G$23 and $G$24; and in geometric group theory it designates the hyperbolic free-splitting graph $G$25. The common theme is a controlled replacement of vertices or splittings of ambient structure, but the mathematical content depends strongly on the context.