---
title: Splitting Graphs in Algebra and Combinatorics
url: https://www.emergentmind.com/topics/splitting-graph
type: topic
---

# Splitting Graphs in Algebra and Combinatorics

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 [1908.03769]. 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
\[
a:V(\widetilde G)\longrightarrow V(G)
\]
such that for every edge \(\{v,w\}\in E(\widetilde G)\), the image \(\{a(v),a(w)\}\) is an edge of \(G\), and the induced map
\[
E(\widetilde G)\longrightarrow E(G),\qquad \{v,w\}\mapsto \{a(v),a(w)\},
\]
is bijective. The map \(a\) is called a splitting map. Intuitively, \(\widetilde G\) “blows up” some vertices of \(G\) into several copies and reattaches edges so that one still sees precisely the same edge-set on the level of \(G\) [1908.03769].

The algebraic setting is built from the edge ideals of the two graphs. If \(G\) has vertex set \(V(G)\), then
\[
I(G)=\bigl(x_i x_j:\{i,j\}\in E(G)\bigr)\subset S=K[x_i:i\in V(G)],
\]
while for \(\widetilde G\) one writes
\[
I(\widetilde G)\subset S'=K[y_v:v\in V(\widetilde G)].
\]
The central problem is then to compare invariants of \(S/I(G)\) and \(S'/I(\widetilde G)\). 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
\[
\operatorname{pd}\bigl(I(G)\bigr)=\operatorname{proj\,dim}_S S/I(G),
\qquad
\operatorname{reg}\bigl(I(G)\bigr)=\operatorname{reg}_S S/I(G).
\]
The guiding expectation is that passing from \(G\) to a splitting graph \(\widetilde G\) should not improve these invariants. More precisely, one hopes always to have
\[
\operatorname{pd}(I(G))\le \operatorname{pd}(I(\widetilde G)),
\]
\[
\operatorname{reg}(I(G))\le \operatorname{reg}(I(\widetilde G)),
\]
and
\[
\beta_i\bigl(I(G)\bigr)\le \beta_i\bigl(I(\widetilde G)\bigr)\qquad \forall\,i.
\]
In full generality these statements remain conjectural [1908.03769].

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 \(a:V(\widetilde G)\to V(G)\) is called special if one of the following holds:

1. Whenever \(v,v'\in V(\widetilde G)\) satisfy \(a(v)=a(v')\), every neighbor of \(v\) in \(\widetilde G\) is adjacent to every neighbor of \(v'\).

2. Whenever \(v\neq v'\) lie in the same fiber \(a(v)=a(v')\), they lie in different connected components of \(\widetilde G\).

Under this hypothesis, Herzog–Moradi–Rahimbeigi prove
\[
\operatorname{pd}(I(G))\le \operatorname{pd}(I(\widetilde G))
\qquad\text{and}\qquad
\operatorname{reg}(I(G))\le \operatorname{reg}(I(\widetilde G)).
\]
The proof reduces a special splitting step-by-step to the merging of two vertices \(x,y\) with the same image, and the main algebraic device is the short exact sequence
\[
0\longrightarrow \frac{S'}{(I:x-y)}(-1)
\xrightarrow{\;\cdot(x-y)\;}
\frac{S'}{I}
\longrightarrow
\frac{S'}{(I,x-y)}
\longrightarrow 0,
\]
where \(I=I(\widetilde G)\subset S'\). In the first special case one shows combinatorially that \((I:x-y)=I\) 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 [1908.03769].

A corollary applies the regularity inequality to several graph classes already known to satisfy \(\operatorname{reg}(I(G))=\nu(G)+1\), where \(\nu(G)\) is the induced-matching number. The listed classes are sequentially Cohen–Macaulay, chordal, weakly chordal, sequentially CM bipartite, unmixed bipartite, very well-covered, and \(C_5\)-free vertex-decomposable graphs. In these cases,
\[
\operatorname{reg}(I(G))=\nu(G)+1
\quad\Longrightarrow\quad
\operatorname{reg}(I(\widetilde G))\ge \nu(G)+1,
\]
and hence \(\operatorname{reg}(I(G))\le \operatorname{reg}(I(\widetilde G))\).

## 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
\[
\beta_i\bigl(I(G)\bigr)\le \beta_i\bigl(I(\widetilde G)\bigr)
\qquad\text{for all }i.
\]
This is the case in which Betti-number monotonicity is proved rather than conjectured [1908.03769].

For arbitrary splitting graphs, one always has the dimension inequality
\[
\dim S'/I(\widetilde G)\ge \dim S/I(G).
\]
If, in addition, \(G\) is a path or an even cycle, then
\[
\depth\, S'/I(\widetilde G)\ge \depth\, S/I(G).
\]
However, depth inequalities fail in general. A counterexample in the same paper shows that splitting may lower depth by at least \(1\), so Cohen–Macaulayness need not be preserved. Accordingly, even when \(G\) is Cohen–Macaulay of one of the special types above, the monotonicity of projective dimension and regularity only says that \(\widetilde G\) has at least as bad projective dimension or regularity; it does not recover Cohen–Macaulayness.

Some purely graph-theoretic properties are preserved. If \(G\) is bipartite, then any splitting graph \(\widetilde G\) is again bipartite. If \(G\) is a forest, then any splitting graph \(\widetilde G\) 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 [1908.03769].

## 5. The classical splitting graph \(S(G)\)

An older graph-theoretic construction uses the notation \(S(G)\). If \(G=(V,E)\) is a finite simple graph of order \(n\), the splitting graph \(S(G)\) is obtained by introducing a new vertex \(v'\) for each \(v\in V\), and joining \(v'\) to \(u\in V\) if and only if \(uv\in E\). Equivalently,
\[
V\bigl(S(G)\bigr)=V\;\dot\cup\;V',
\qquad
E\bigl(S(G)\bigr)=E\;\dot\cup\;E',
\]
where \(V'=\{v':v\in V\}\) and
\[
E'=\{\,u\,v':uv\in E\}.
\]
Here \(|V'|=n\) and \(|E'|=2|E|\) [2602.20504].

In Sampathkumar–Walikar’s 1980 treatment, it was asserted that for any \(G\) of order \(n\),
\[
\alpha_0\bigl(S(G)\bigr)=n
\qquad\text{and}\qquad
\beta_0\bigl(S(G)\bigr)=n,
\]
where \(\alpha_0(G)\) is the vertex-cover number and \(\beta_0(G)\) 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
\[
\beta_0^*(G)=\max_{I\subseteq V\text{ independent}}\bigl(|I|-|N_G(I)|\bigr),
\]
they prove that if \(G\) is a connected simple graph of order \(n\ge 2\), then
\[
\beta_0\bigl(S(G)\bigr)=n+\beta_0^*(G),
\qquad
\alpha_0\bigl(S(G)\bigr)=n-\beta_0^*(G).
\]
Equivalently, the old formulas hold precisely when \(\beta_0^*(G)=0\), that is, precisely when \(|N_G(I)|\ge |I|\) for every independent set \(I\subseteq V\) [2602.20504].

The corrected formulas imply the sharp range
\[
n\le \beta_0\bigl(S(G)\bigr)\le 2n-2,
\]
and every integer in that interval is attained by a suitable graph of order \(n\). 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 \(n\).

## 6. Terminological scope and adjacent research directions

The literature uses closely related names for several distinct constructions.

| Term | Definition | Representative result |
|---|---|---|
| Splitting graph of \(G\) | A graph \(\widetilde G\) with a surjection \(a:V(\widetilde G)\to V(G)\) whose induced edge map is bijective | Comparison of \(\operatorname{pd}\), \(\operatorname{reg}\), Betti numbers, depth, and dimension [1908.03769] |
| Classical splitting graph \(S(G)\) | Add one vertex \(v'\) for each \(v\in V(G)\), with \(v'\) adjacent to exactly the neighbors of \(v\) in \(G\) | \(\beta_0(S(G))=n+\beta_0^*(G)\), \(\alpha_0(S(G))=n-\beta_0^*(G)\) [2602.20504] |
| \(m\)-splitting graph \(\mathcal S_m(G)\) | Add \(m\) new copies \(u_i^k\) of each vertex-neighborhood pattern | \(E(\mathcal S_m(G))=\sqrt{1+4m}\,E(G)\) [2604.00040] |
| Free-splitting graph \(FS_n\) | Vertices are conjugacy classes of one-edge free splittings of \(F_n\) | \(FS_n\) is Gromov hyperbolic and \(\partial FS_n\cong S\mathcal T/\!\sim\) [1211.1630] |

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 \(2K_2\), \(C_4\), and \(C_5\), and they further divide into balanced and unbalanced classes [1506.03746].

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 \(k\) 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 [2602.04628]. For plane graphs, the outerplane splitting number \(\sigma(G)=k\) is equivalent to the existence of a connected face cover of size \(k+1\), and also equivalent to a minimum feedback vertex set of size \(k+1\) in the dual; the associated decision problem is NP-complete for plane biconnected graphs, but maximal planar graphs admit a polynomial-time algorithm [2301.09440]. 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 \(2^{O(k^2)}\cdot n^{O(1)}\) [2202.12293].

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 \(S(G)\); in spectral graph theory it includes \(\mathcal S_m(G)\) and \(\mathcal S_{p,q}(G)\); and in geometric group theory it designates the hyperbolic free-splitting graph \(FS_n\). The common theme is a controlled replacement of vertices or splittings of ambient structure, but the mathematical content depends strongly on the context.

Source: https://www.emergentmind.com/topics/splitting-graph