About a counterexample on contractible transformations of graphs

TL;DR

This paper critically examines Ivashchenko's claims about contractible transformations of graphs, providing a counterexample that invalidates his previous results and highlighting errors in his proofs.

Contribution

The paper presents a counterexample demonstrating that Ivashchenko's earlier results on contractible transformations are incorrect, correcting the understanding of graph homology invariance.

Findings

Counterexample disproves Ivashchenko's claims

Identifies errors in previous proofs

Clarifies the limitations of contractible transformations

Abstract

In [A. V. Ivashchenko, Contractible transformations do not change the homology groups of graphs, Discrete Mathematics 126 (1) (1994) p 159,170], Ivashchenko started with the study of contractible graphs, he began with this because they have application to molecular spaces. But in a second paper [A. V. Ivashchenko, Some properties of contractible transformations on graphs, Discrete Mathematics 133 (1) (1994) p 139,145], he made several mistakes. We show in this paper that the results in that second paper are false or the proofs are wrong.

Full text

††journal: Discrete Mathematics

1 Introduction

In graph theory, several reductions that leave the homology invariant have been studied. In [1, 2], A. Ivashchenko shows a family of graphs constructed from $K(1)$ by contractible transformations (as in Definition 1), and he proves that such transformations do not change the homology groups of graphs. He started the study of these transformations because these are used in the theory of molecular spaces and digital topology. Modern references are [3, 4, 5]. Particulary [5] contributes with

an aplication to topological data analisys, 2. 2.

they have proved that the Ivashchenko’s contractible graphs are collapsibles 3. 3.

and we can also see as a collorary from the main result of [1] i.e. that the homology do not change by the Ivashchenko’s transformations.

We show in this paper that most of the results in [2] are false or the proofs are wrong.

2 Contractible transformations

Let $G=(V(G),E(G))\in\mathfrak{G}$ be a graph, and let $v\in V(G)$ be a vertex. We denote $N_{G}(v)=\{u\in V(G):\{u,v\}\in E(G)\}$ and $N_{G}(v,w)=N_{G}(v)\cap N_{G}(w)$. In addition, by abuse of notation we identify the graph with its set of vertices.

In [1] the next family of graphs was defined, and its elements are called contractible graphs.

Definition 1

Let $\mathfrak{I}\subset\mathfrak{G}$ be the family of graphs defined by

The trivial graph $K(1)$ is in $\mathfrak{I}$. 2. 2.

Any graph of $\mathfrak{I}$ can be obtained from $K(1)$ by the following transformations.

1. (I1)

Deleting a vertex $v$. A vertex $v$ of a graph $G$ can be deleted if $N_{G}(v)\in\mathfrak{I}$. 2. (I2)

Gluing a vertex $v$. If a subgraph $G_{1}$ of the graph $G$ is in $\mathfrak{I}$, then the vertex $v$ can be glued to the graph $G$ in such way that $N_{G}(v)=G_{1}$. 3. (I3)

Deleting an edge $\{v_{1},v_{2}\}$. The edge $\{v_{1},v_{2}\}$ of a graph $G$ can be deleted if $N_{G}(v_{1},v_{2})\in\mathfrak{I}$. 4. (I4)

Gluing an edge $\{v_{1},v_{2}\}$. Let two vertices $v_{1}$ and $v_{2}$ of a graph $G$ be nonadjacent. The edge $\{v_{1},v_{2}\}$ can be glued if $N_{G}(v_{1},v_{2})\in\mathfrak{I}$.

If $G$ belongs to $\mathfrak{I}$, then $G$ is called a contractible graph.

The transformations (I1)-(I4) were referred in [1] as contractible transformations. The contractible transformations are used in molecular spaces, see [1] for more explanation. In addition, in [1] it was proved that contractible transformations do not change the homology groups of a graph, for any commutative group of coefficients $A$, so the elements of $\mathfrak{I}$ have trivial groups of $A$-homology.

In [1, Th. 4.9], it was proved that the contractible transformation does not change the homology groups of the graph $G$ when $N_{G}(v)\in\mathfrak{I}$.

3 Counterexample

We cite textualy the Axiom 3.4 at [2]

Axiom 2

Suppose that $G$ is a contractible graph, and a vertex $v$, $v\in G$, is not adjacent to some vertices of $G$. Then there exists a nonadjacent vertex $u$, $u\in G$, such that the subgraph $O(vu)$ is contractible.

Where $O(vu)$ is the induce graph by $N(v)\cap N(u)$.

Ivashchenko claimed that the previous axiom is verified on small graphs and he did not intent to prove the generic case. But in the heart graph of figure 1, we can see that vertex $1$ is not adjecent to $8$, $9$, $11$ y $12$, we can see that common neighboorhood of $1$ with each of those vertices is not contractible. So the Ivashchenko’s axiom is false. All the results of that paper are based in the axiom 3.4, and so the Theorems 3.5, 3.8, 3.9, 3.10 and Corollary 3 are clearly false, the heart graph in figure 1 shows this. For example theorem 3.5 establishes that any contractible graph has two contractible vertices and in figure 1 we clearly see that the heart graph is contractible and it does not have any contractible vertex. We think that theorem 3.7, 3.11 and 3.12 are true, but the proofs are incorrect because they use the axiom 3.4 or some of its consecuenses. The only theorem that is correctly proved it is 3.13.

The heart graph in figure 1 is the smallest graph visually pleasing that we found. We are wondering if there is another with less vertices.

Acknowledgment

I thank Anton Dochthermann, Jesús Espinoza, Etiene Fieux and Héctor Hernández for their friendly and solidary support in the writing of this note.