Digraphs with at most one trivial critical ideal

TL;DR

This paper characterizes digraphs with at most one trivial critical ideal, linking properties of their critical groups, Smith groups, and characteristic polynomials, thus extending understanding of algebraic invariants of digraphs.

Contribution

It provides a complete characterization of digraphs with at most one trivial critical ideal, connecting critical ideals with critical and Smith groups.

Findings

Characterization of digraphs with at most one trivial critical ideal

Implications for critical groups with a single invariant factor of one

Implications for Smith groups with a single invariant factor of one

Abstract

Critical ideals generalize the critical group, Smith group and the characteristic polynomials of the adjacency and Laplacian matrices of a graph. We give a complete characterization of the digraphs with at most one trivial critical ideal. Which implies the characterizations of the digraphs whose critical group has one invariant factor equal to one, and the digraphs whose Smith group has one invariant factor equal to one.

Tables

Table 1. Table 1. The number of γ 𝛾 \gamma -critical digraphs with n 𝑛 n vertices and algebraic co-rank k 𝑘 k .

$k\backslash n$	2	3	4	5	6
1	2
2		7	10
3			61	1308	414
4				1183	542437
5					38229

Equations

$$L(D,X_D{)}_{uv}=\{\begin{array}{ll}{x}_u& \text{ if }u=v,\\ -m_{uv}& \text{ otherwise},\end{array}$$

$$I_i(D,X_D)=⟨\{\det (m):m\text{ is an }i\times i\text{ submatrix of }L(D,X_D)\}⟩\subseteq \mathbb{Z}[X_D].$$

$${\Gamma }_{\le i}=\{D:D\text{ is a simple connected digraph with }\gamma (D)\le i\}.$$

$$F_{3,1}$$

$$F_{3,6}$$

$$F_{4,6}$$

$$I_2({\Lambda }_{n_1,n_2,n_3},\{X_T,Y_K,Z_{T^{\prime }}\})=\{\begin{array}{ll}⟨{\cup }_{i=1}^{n_1}{x}_i,{\cup }_{i=1}^{n_2}(y_i+1),{\cup }_{i=1}^{n_3}{z}_i⟩,& \text{ if }{n}_1,n_2,n_3\ge 1,\\ ⟨x_1{y}_1⟩,& \text{ if }{n}_1=n_2=1,n_3=0\\ ⟨x_1{z}_1⟩,& \text{ if }{n}_1=n_3=1,n_2=0\\ ⟨y_1{z}_1⟩,& \text{ if }{n}_2=n_3=1,n_1=0\\ ⟨{\cup }_{i=1}^{n_1}{x}_i⟩,& \text{ if }{n}_1\ge 2,n_2=1,n_3=0,\\ ⟨{\cup }_{i=1}^{n_1}{x}_i,{\cup }_{i=1}^{n_2}(y_i+1)⟩,& \text{ if }{n}_1\ge 1,n_2\ge 2,n_3=0,\\ ⟨y_1{y}_2-1⟩,& \text{ if }{n}_1=0,n_2=2,n_3=0,\\ ⟨{\cup }_{i<j}(y_i+1)(y_j+1)⟩,& \text{ if }{n}_1=0,n_2\ge 3,n_3=0,\\ ⟨{\cup }_{i=1}^{n_1}{z}_i⟩,& \text{ if }{n}_1=0,n_2=1,n_3\ge 2,\\ ⟨{\cup }_{i=1}^{n_2}(y_i+1),{\cup }_{i=1}^{n_3}{z}_i⟩,& \text{ if }{n}_1=0,n_2\ge 2,n_3\ge 1,\\ ⟨{\cup }_{i=1}^{n_3}{z}_i⟩,& \text{ if }{n}_1=1,n_2=0,n_3\ge 2,\\ ⟨{\cup }_{i=1}^{n_1}{x}_i⟩,& \text{ if }{n}_1\ge 2,n_2=0,n_3=1,\\ ⟨{\cup }_{i=1}^{n_1}{x}_i,{\cup }_{i=1}^{n_3}{z}_i⟩,& \text{ if }{n}_1\ge 2,n_2=0,n_3\ge 2.\end{array}$$

$$L(\Lambda_{n_{1},n_{2},n_{3}},\{X_{T},Y_{K},Z_{T^{\prime}}\})=\left[\begin{array}[]{ccc}L(T,X_{T})&-J_{n_{1},n_{2}}&-J_{n_{1},n_{3}}\\ \bf{0}_{n_{2},n_{1}}&L(K,Y_{K})&-J_{n_{2},n_{3}}\\ \bf{0}_{n_{3},n_{1}}&\bf{0}_{n_{3},n_{2}}&L(T^{\prime},Z_{T^{\prime}})\end{array}\right],$$

$$A_1$$

$$I_i(D,X_D){|}_{X_D={\mathrm{deg}}^{+}(D)}=⟨\prod _{j=1}^i{f}_j⟩=⟨{\Delta }_i(L(D))⟩\text{ for all }1\le i\le n-1.$$

$${\mathcal{G}}_i=\{D:D\text{ is a simple connected digraph with }{f}_1(D)=i\}.$$

$${\mathrm{deg}}^{+}(v)=\{\begin{array}{ll}{n}_2+n_3& \text{if }v\in T_{n_1},\\ n_2+n_3-1& \text{if }v\in K_{n_2},\\ 0& \text{if }v\in T_{n_3}^{\prime }.\end{array}$$

$$I_i(D,X_D){|}_{X_D=0}=⟨\prod _{j=1}^i{f}_j⟩=⟨{\Delta }_i(A(D))⟩\text{ for all }1\le i\le n-1.$$

Full text

The authors were partially supported by SNI

1. Introduction

In the following, the underlying graph of any digraph must be connected, and will contain no loops. Given a digraph $D=(V,A)$ and a set of indeterminates $X_{D}=\{x_{u}\,:\,u\in V(D)\}$, the generalized Laplacian matrix $L(D,X_{D})$ of $D$ is the matrix with rows and columns indexed by the vertices of $D$ given by

[TABLE]

where $m_{uv}$ is the number of arcs going from $u$ to $v$.

Definition 1.1**.**

For all $1\leq i\leq|V|$, the $i$-th critical ideal of $D$ is the determinantal ideal given by

[TABLE]

We say that a critical ideal $I_{i}(D,X_{D})$ is trivial when it is equal to $\langle 1\rangle$.

Definition 1.2**.**

The algebraic co-rank $\gamma(D)$ of a digraph $D$ is the number of trivial critical ideals of $D$.

Most of the basic properties of the critical ideals were obtained in [8]. For instance, it was proven that if $H$ is an induced subdigraph of $G$, then $I_{i}(H,X_{H})\subseteq I_{i}(G,X_{G})$ for all $i\leq|V(H)|$. Thus $\gamma(H)\leq\gamma(G)$. In this work, we are interested in describing the following digraph family:

Definition 1.3**.**

For $i\in\mathbb{N}$, let

[TABLE]

Our main result is the characterization of $\Gamma_{\leq 1}$. This implies two characterizations: the digraphs whose critical group has one invariant factor equal to 1, and the digraphs whose Smith group has one invariant factor equal to one. These characterizations generalize, in the digraph context, the characterization of the connected graphs with one trivial critical ideal and the characterization of connected graphs with one invariant factor equal to 1. For this, we will study in Section 2 the minimal forbidden digraphs for $\Gamma_{\leq i}$. These digraphs have been playing a crucial role in the understating the critical ideals and their classification, and allow us to give the characterization of the digraphs with one trivial critical ideal. In Section 3, we will give the complete characterization of $\Gamma_{\leq 1}$. Finally, in Section 4, we give the characterizations of the digraphs whose critical group has one invariant factor equal to 1, and the digraphs whose Smith group has one invariant factor equal to one.

2. Forbidden and $\gamma$-critical digraphs

The major advantage of the critical ideals over the critical group and the Smith group is that critical ideals behave well under induced subdigraph property. This property allow us to define the following concepts.

Definition 2.1**.**

We say that a digraph $D$ is forbidden for $\Gamma_{\leq k}$ if and only if $\gamma(D)\geq k+1$. Moreover, let ${\bf Forb}(\Gamma_{\leq k})$ be the set of minimal (under induced subdigraphs property) forbidden digraphs for $\Gamma_{\leq k}$.

Given a family $\mathcal{F}$ of digraphs, a digraph $D$ is called $\mathcal{F}$-free if no induced subdigraph of $D$ is isomorphic to a member of $\mathcal{F}$. Thus $D\in\Gamma_{\leq k}$ if and only if $D$ is ${\bf Forb}(\Gamma_{\leq k})$-free. And equivalently, $D$ belongs to $\Gamma_{\geq k+1}$ if and only if $D$ contains a digraph of ${\bf Forb}(\Gamma_{\leq k})$ as an induced subgraph. Hence a characterization of ${\bf Forb}(\Gamma_{\leq k})$ leads to a characterization of $\Gamma_{\leq k}$.

As $k$ grows, it becames difficult to give a complete description of ${\bf Forb}(\Gamma_{\leq k})$. An alternative technique of computing the elements of ${\bf Forb}(\Gamma_{\leq k})$ is by means of the following definition. A digraph $D$ is called $\gamma$-critical if $\gamma(D\setminus v)<\gamma(D)$ for all $v\in V(D)$. That is, $D\in{\bf Forb}(\Gamma_{\leq k})$ if and only if $D$ is $\gamma$-critical such that $\gamma(D)\geq k+1$ and $\gamma(D-v)\leq k$. We implemented this criterion in the software Sage [21] and Nauty [15], Table 1 shows the number of $\gamma$-critical digraphs with at most 6 vertices.

Using the data obtained from this computation, we have that the directed path $\overrightarrow{P_{2}}$ with 2 vertices and the directed cycle $\overrightarrow{C_{2}}$ with 2 vertices are the minimal forbidden digraphs for $\Gamma_{\leq 0}$, see Fig. 1. Since any other connected digraph with more than $2$ vertices contains $\overrightarrow{P_{2}}$ or $\overrightarrow{C_{2}}$ as induced digraphs, then ${\bf Forb}(\Gamma_{\leq 0})=\{\overrightarrow{P_{2}},\overrightarrow{C_{2}}\}$. Therefore, the digraph $T_{1}$ consisting of an unique vertex is the only $(\overrightarrow{P_{2}},\overrightarrow{C_{2}})$-free connected digraph, that is, $\Gamma_{\leq 0}=\{T_{1}\}$.

Also, we have that ${\bf Forb}(\Gamma_{\leq 1})$ consists of $7$ digraphs with $3$ vertices and $10$ digraphs with $4$ vertices (see Figure 2). These digraphs will be used to completely characterize $\Gamma_{\leq 1}$ in the next section.

Note that the set of digraphs in ${\bf Forb}(\Gamma_{\leq 2})$ is bigger. For instance, at least contains $61$ digraphs with $4$ vertices, $1308$ digraphs with $5$ vertices, and $414$ digraphs with $6$ vertices. The size and complexity of ${\bf Forb}(\Gamma_{\leq k})$, for $k\geq 2$, make difficult to obtain a complete characterization of $\Gamma_{\leq k}$.

3. Digraphs with one trivial critical ideal

The main goal of this section is to give the characterization of the digraphs with at most one trivial critical ideal. As in the case of simple graphs and digraphs with algebraic co-rank equal to zero, the characterization of $\Gamma_{\leq 1}$ relies in that $\Gamma_{\leq 1}$ is closed under induced subdigraphs and that we have previously computed the $\gamma$-critical digraphs with algebraic co-rank equal to $2$.

Let $\mathfrak{F}$ be the family of digraphs shown in Figure 2; consisting of $17$ $\gamma$-critical digraphs with algebraic co-rank equal to $2$. In appendix A there is a sage code to verify the following lemma.

Lemma 3.1**.**

If $\mathfrak{F}$ is the family of digraphs shown in Figure 2, then $\mathfrak{F}\subseteq{\bf Forb}(\Gamma_{\leq 1})$.

A digraph $D$ is complete if, for every pair $u,v$ of distinct vertices of $D$, both arcs $(uv)$ and $(vu)$ are in $D$. For simplicity, let $uv$ denote the existence of both arcs $(uv)$ and $(vu)$. The complete digraph with $n$ vertices is denoted by ${K_{n}}$. The trivial digraph with $n$ vertices with no arcs is denoted by $T_{n}$. Let $D_{1}$ and $D_{2}$ be vertex-disjoint subdigraphs of $D$. The set of arcs with tails in $V(D_{1})$ and heads in $V(D_{2})$ is denoted by $(D_{1},D_{2})_{D}$. We say that $(D_{1},D_{2})_{D}$ is complete when is equal to $\{uv\,:\,u\in V(D_{1})\text{ and }v\in V(D_{2})\}$.

Let $\Lambda_{n_{1},n_{2}n_{3}}$ be the digraph defined in the following way: The vertex set $V(\Lambda_{n_{1},n_{2}n_{3}})$ is partitioned in three sets $T$, $T^{\prime}$ and $K$ with $n_{1},n_{3}$ and $n_{2}$ vertices, respectively, such that $T$ and $T^{\prime}$ are two trivial digraphs, and $K$ is a complete digraph. Additionally, the arc sets $(T,K)_{\Lambda_{n_{1},n_{2},n_{3}}}$, $(T,T^{\prime})_{\Lambda_{n_{1},n_{2},n_{3}}}$ and $({K},T^{\prime})_{\Lambda_{n_{1},n_{2},n_{3}}}$ are complete. See Figure 3 for a graphical representation of $\Lambda_{n_{1},n_{2}n_{3}}$. Let $\Lambda$ be the family of digraphs consisting of all the connected digraphs $\Lambda_{n_{1},n_{2}n_{3}}$ for all $n_{1},n_{2},n_{3}\geq 0$.

Lemma 3.2**.**

If $\Lambda_{n_{1},n_{2},n_{3}}$ is a connected digraph with $n_{1}+n_{2}+n_{3}\geq 2$, then $I_{1}(\Lambda_{n_{1},n_{2},n_{3}},\{X_{T},Y_{K},Z_{T^{\prime}}\})$ is trivial, and

[TABLE]

Proof.

We have that

[TABLE]

where $J_{m,n}$ denote the all ones $m\times n$-matrix. It is easy to see that the first critical ideal of a digraph with at least one arc is trivial. For the case when $n_{1},n_{2},n_{3}\geq 1$, we have that the non-singular 2-minors (with positive leading coefficient) of the generalized Laplacian matrix of $\Delta_{n_{1},n_{2},n_{3}}$ are of the form: $x_{i}x_{j}$, $x_{i}$, $x_{i}y_{j}$, $y_{i}+1$, $y_{i}z_{j}$, $x_{i}z_{j}$, $z_{i}$, $y_{i}y_{j}-1$, $z_{i}z_{j}$. Since $x_{i}x_{j}$, $x_{i}y_{j}$, $y_{i}z_{j}$, $x_{i}z_{j}$ and $y_{i}y_{j}-1$ are in $\left<\cup_{i=1}^{n_{1}}x_{i},\cup_{i=1}^{n_{2}}(y_{i}+1),\cup_{i=1}^{n_{3}}z_{i}\right>$, we have that $L(\Lambda_{n_{1},n_{2},n_{3}},\{X_{T},Y_{K},Z_{T^{\prime}}\})=\left<\cup_{i=1}^{n_{1}}x_{i},\cup_{i=1}^{n_{2}}(y_{i}+1),\cup_{i=1}^{n_{3}}z_{i}\right>$. The non-singular 2-minors of the rest of the cases are a subset of the obtained previously, from which the result follows. ∎

Now we give the characterization of the digraphs with at most one trivial critical ideal.

Theorem 3.3**.**

If $D$ is a connected simple digraph, then the following statements are equivalent:

1. i.

$D\in\Gamma_{\leq 1}$, 2. ii.

$D$* is $\mathfrak{F}$-free.* 3. iii.

$D$* is isomorphic to $\Lambda_{n_{1},n_{2},n_{3}}$ for some natural numbers $n_{1}$, $n_{2}$ and $n_{3}$.*

Proof.

By Lemma 3.1, each digraph in $\mathfrak{F}$ has algebraic co-rank 2, then a digraph in $\Gamma_{\leq 1}$ is $\mathfrak{F}$-free. Also, Lemma 3.2 implies that each digraph in $\Lambda$ belongs to $\Gamma_{\leq 1}$. Therefore, it remains to prove that a $\mathfrak{F}$-free digraph is isomorphic to $\Lambda_{n_{1},n_{2},n_{3}}$ for some integers $n_{1}$, $n_{2}$ and $n_{3}$. We will proceed by induction on $n=n_{1}+n_{2}+n_{3}$.

We have that $\overrightarrow{P_{2}}$ and $\overrightarrow{C_{2}}$ are the only digraphs with 2 vertices with algebraic co-rank equal to 1, and in Figure 4 there are shown the only six digraphs with algebraic co-rank equal to 1. The result is true for $n\leq 3$, since all these digraphs belong to $\Lambda$.

Assume that, for some integer $n\geq 4$, every $\mathfrak{F}$-free digraph on $n$ vertices is isomorphic to $\Lambda_{n_{1},n_{2},n_{3}}$ for some $n_{1}$, $n_{2}$ and $n_{3}$. Now let $D=(V,A)$ be an $\mathfrak{F}$-free digraph on $n+1$ vertices, and let $v\in V(D)$. The digraph $D^{\prime}=D-v$ is $\mathfrak{F}$-free, and by inductive hypothesis, $D^{\prime}$ is isomorphic to $\Lambda_{n_{1},n_{2},n_{3}}$ for some integers $n_{1}$, $n_{2}$ and $n_{3}$. Take three vertices $a,b,c$ in $D^{\prime}$ such that its underlying graph is connected, and $v$ is adjacent with at least one of the three vertices. Since $D^{\prime}$ is isomorphic to $\Lambda$, the induced digraph by the vertices $a,b$ and $c$ is isomorphic to one of the digraphs shown in Figure 4. We are going to continue with the following claims.

Claim 3.4**.**

If $D^{\prime}$ is a complete digraph, then

1. i.

each vertex in $D^{\prime}$ is adjacent to $v$, 2. ii.

all arcs between $D^{\prime}$ and $v$ have the same orientation.

Proof.

Since $D$ is connected, then $v$ is adjacent to a vertex $a$ in $D^{\prime}$. Take $b\in V(D^{\prime})$ different from $a$. Suppose $b$ is not adjacent with $v$, then since there are three possibilities in which $a$ and $v$ are adjacent: either $(av)$, $(va)$, or $av$, that is, both $(av)$ and $(va)$ exist. However, these possibilities cannot be, since the induced digraph by $a,b$ and $v$ would be isomorphic to $F_{3,2}$, $F_{3,3}$ or $F_{3,4}$. Thus, each vertex in $D^{\prime}$ is adjacent to $v$. Similarly, all arcs between $D^{\prime}$ and $v$ have the same orientation, otherwise $D^{\prime}$ would contain an induced subgraph isomorphic to $F_{3,6}$ or $F_{3,7}$. ∎

Remark 3.5**.**

Let $a,b,c\in D^{\prime}$ such that $(ab),(bc)\in A(D^{\prime})$ and $(ba),(cb)\notin A(D^{\prime})$. If $a$ and $c$ are adjacent, then $(ac)$ is the only possible arc between both vertices.

Claim 3.6**.**

Assume that the unique arcs in $D^{\prime}[a,b,c]$ are $(ac)$ and $(bc)$. Then, $(\{a,b,c\},v)$ is equal to one of the following arcs sets.

1. i.

$\{(av),(bv)\}$, 2. ii.

$\{(av),(bv),(vc)\}$, 3. iii.

$\{(av),(bv),(cv)\}$, 4. iv.

$\{(av),(bv),vc\}$, 5. v.

$\{(vc)\}$.

Proof.

Suppose $(va)$ exists. Then, by Remark 3.5, $(vc)$ exits. There must exists an arc between $b$ and $v$, otherwise $D^{\prime}[a,b,c,v]$ would be isomorphic to $F_{4,2}$. The arcs $vb$ or $(bv)$ cannot exist, otherwise $(ba)$ should exits. Finally, it is impossible that the arc $(vb)$ exits, otherwise $D^{\prime}[a,b,c,v]$ would be isomorphic to $F_{4,6}$. Thus, $(va)\notin E(D^{\prime})$.

Now, suppose $va\in E(D^{\prime})$. By Claim 3.4, $(vc)\in E(D^{\prime})$. And, there should exists an arc between $v$ and $b$, otherwise $D^{\prime}[a,b,c,v]$ would be isomorphic to $F_{4,4}$. But, since $va$ is complete, then by Claim 3.4 there is an arc between $v$ and $b$ if and only if there is an arc between $a$ and $b$; which is impossible. Then, $va\notin E(D^{\prime})$.

Suppose $(av)$ exists. Neither $(vb)$ or $vb$ exists, otherwise $(ab)$ should exists; which is not possible. Considering $(bv)\in E(V^{\prime})$, the cases $i$, $ii$, $iii$ and $iv$ are yielded.

The cases between $v$ and $b$ can be analyzed similarly. Now, we assume there exists no arc between $v$ and the vertices $a$ and $b$. There is not possible that $(cv)$ or $cv$ exist, otherwise $(av)$ and $(bv)$ should exist. The last case is that $vc\in E(D^{\prime})$, which corresponds to case $v$. ∎

Claim 3.7**.**

Assume that the unique arcs in $D^{\prime}[a,b,c]$ are $(ca)$ and $(cb)$. Then, $(\{a,b,c\},v)$ is equal to one of the following arcs sets.

1. i.

$\{(va),(vb)\}$, 2. ii.

$\{(va),(vb),(vc)\}$, 3. iii.

$\{(va),(vb),(cv)\}$, 4. iv.

$\{(va),(vb),vc\}$, 5. v.

$\{(cv)\}$.

Proof.

Suppose $(va)$ exists. There must exists an arc between $v$ and one of the vertices $b$ or $c$, otherwise $D^{\prime}[a,b,c,v]$ would be isomorphic to $F_{4,1}$. Considering $(vb)\in E(V^{\prime})$, the cases $i$, $ii$, $iii$ and $iv$ are yielded. On the other hand, the arcs $vb$ or $(bv)$ cannot exist, otherwise $(ba)$ should exits.

Suppose $va\in E(D^{\prime})$. By Claim 3.4, $(cv)\in E(D^{\prime})$. And, there should exists an arc between $v$ and $b$, otherwise $D^{\prime}[a,b,c,v]$ would be isomorphic to $F_{4,5}$. On the other hand, there is no arc between $v$ and $b$, because there should exist an arc between $a$ and $b$; which is impossible. Then, $va\notin E(D^{\prime})$.

Suppose $(av)$ exists. Then, by Remark 3.5, $(cv)\in E(D^{\prime})$. There should exists an arc between $v$ and $b$, otherwise $D^{\prime}[a,b,c,v]$ would be isomorphic to $F_{4,3}$. Neither $(vb)$ or $vb$ exists, otherwise $(ba)$ should exists; which is not possible. Considering $(bv)\in E(V^{\prime})$, $D^{\prime}[a,b,c,v]$ would be isomorphic to $F_{4,6}$; which is impossible. Thus, $(av)\notin E(D^{\prime})$.

The cases between $v$ and $b$ can be analyzed similarly. Now, suppose there exists no arc between $v$ and the vertices $a$ and $b$. There is not possible that any of the arcs $(vc)$ or $vc$ exist, otherwise $(va)$ and $(vb)$ should exist. Finally, $cv\in E(D^{\prime})$, which corresponds to case $v$. ∎

Claim 3.8**.**

Assume that the unique arcs in $D^{\prime}[a,b,c]$ are $(ab)$, $(ac)$ and $(bc)$. Then, $(\{a,b,c\},v)$ is equal to one of the following arcs sets.

1. i.

$\{(av),vb,(vc)\}$, 2. ii.

$\{(vb),(vc)\}$, 3. iii.

$\{(av),(bv)\}$.

Proof.

The arcs $(va)$ or $va$ are not in $E(D^{\prime})$. Since otherwise $(vb)$ and $vc$ should exist, then $D^{\prime}[a,b,c,v]$ would be isomorphic to $F_{4,7}$ and $F_{4,8}$, respectively. Which is impossible.

Suppose $(av)$ exists. Then, there must exists an arc between $v$ and one of the vertices $b$ or $c$, otherwise $D^{\prime}[a,b,c,v]$ would be isomorphic to $F_{4,3}$, which is forbidden. The arc $(vb)$ cannot exists, since otherwise it would imply that the arc $(vc)$ should exists, but in this case $D^{\prime}[a,b,c,v]$ would be isomorphic to $F_{4,7}$; which is not possible. In the case when $(bv)\in E(V^{\prime})$, we have 4 possibilities for $v$ and $c$: $(vc)$, $vc$, $(cv)$ or $v$ is not adjacent with $c$. The arcs $(vc)$, $vc$ and $(cv)$ are not possible, since otherwise $D^{\prime}[a,b,c,v]$ would be isomorphic to $F_{4,7}$, $F_{4,10}$ and $F_{4,7}$, respectively. In the case when $v$ is not adjacent with $c$, the arc set $(\{a,b,c\},v)$ corresponds to case $iii$. Finally, when the arc $vb$ exists, the arc $(vc)$ exists, and this corresponds to case $i$.

Now, suppose there is no arc between $v$ and $a$. If $(vb)\in E(V^{\prime})$, then $(vc)\in E(V^{\prime})$ and this case corresponds to $ii$. The cases when the arcs $(bv)$ or $vb$ exist, there must exists an arc between $a$ and $v$, which is not the case, therefore these cases are not possible.

Now, suppose there is no arc between $v$ and the vertices $a$ and $b$. The three possibilities for the arc between $v$ and $c$ are not possible, since otherwise $F_{4,2}$, $F_{3,1}$ or $F_{3,2}$ would appear. ∎

Claim 3.9**.**

Assume that the unique arcs in $D^{\prime}[a,b,c]$ are $ab$, $(ac)$ and $(bc)$. Then, $(\{a,b,c\},v)$ is equal to one of the following arcs sets.

1. i.

$\{(av),(bv)\}$, 2. ii.

$\{av,bv,(vc)\}$, 3. iii.

$\{(va),(vb),(vc)\}$.

Proof.

Suppose $(av)\in E(V^{\prime})$. Then, $(bv)$ also is in $E(V^{\prime})$. And there is no arc between $v$ and $c$, otherwise $D^{\prime}[a,b,c,v]$ would be isomorphic to $F_{4,8}$ or $F_{4,9}$. Now, suppose $av\in E(V^{\prime})$. Then, $bv$ and $vc$ are also in $E(V^{\prime})$, and this case corresponds to $ii$. Finally, suppose $(va)\in E(V^{\prime})$. Then, Claim 3.4 implies that $(vb)\in E(V^{\prime})$, and by Remark 3.5 $(vc)$ also is in $E(V^{\prime})$. This case corresponds to $iii$.

The cases where $v$ and $b$ are adjacent can be analyzed similarly to the previous cases. Then, let us continue with the case where there are no arcs between $v$ and the vertices $a$ and $b$. That is, when only one of the arcs $(vc)$, $vc$ or $cv$ exists. These cases are impossible, since otherwise $D^{\prime}[a,b,c,v]$ would be isomorphic to $F_{4,4}$, $F_{3,2}$ or $F_{3,1}$, respectively. ∎

Claim 3.10**.**

Assume that the unique arcs in $D^{\prime}[a,b,c]$ are $ab$, $(ca)$ and $(cb)$. Then, $(\{a,b,c\},v)$ is equal to one of the following arcs sets.

1. i.

$\{(va),(vb)\}$, 2. ii.

$\{av,bv,(cv)\}$, 3. iii.

$\{(av),(bv),(cv)\}$.

Proof.

Note that if $(va)$ or $(vb)$ exists, then by Claim 3.4 both arcs must exist. And there is no arc between $v$ and $c$, since otherwise $D^{\prime}[a,b,c,v]$ would be isomorphic to $F_{4,9}$, or $F_{4,10}$. Thus, $(\{a,b,c\},v)$ is equal to case $i$. If $va$ or $vb$ exists, then by Claim 3.4 both arcs must exist. And by the same claim, $cv$ is in $E(V^{\prime})$. Then, $(\{a,b,c\},v)$ is equal to case $ii$. Finally, if $(av)$ or $(bv)$ exists, then by Claim 3.4 both arcs must exist. And by Remark 3.5 $(cv)$ also is in $E(V^{\prime})$. Therefore, $(\{a,b,c\},v)$ is equal to case $iii$.

Now, let us assume there are no arcs between $v$ and the vertices $a$ and $b$, and only one of the arcs $(vc)$, $vc$ or $cv$ exists. The cases when one of the arcs $vc$ or $cv$ exists are impossible, since otherwise $D^{\prime}[a,b,c,v]$ would be isomorphic to $F_{3,1}$ or $F_{4,5}$, respectively. And finally, $vc$ cannot exist, since otherwise by Claim 3.4 $v$ should be adjacent to $a$ and $b$. ∎

By the previous claims, $v$ must be included into one of the three partitions of $D^{\prime}$: $T$, $T^{\prime}$ or $K$. Therefore, $D$ belongs is isomorphic to a digraph in $\Lambda$. ∎

4. Applications to the Critical group and the Smith group

Originally, critical ideals were defined as a generalization of the critical group, see [2, 3, 8]. However, it is also a generalization of several other algebraic objects like Smith group or characteristic polynomials of the adjacency and Laplacian matrices.

The characterization of the family of simple connected (di)graphs having critical group with $i$ invariant factors equal to $1$ has been of great interest. Probably, it was initially posed by R. Cori111Personal communication with C. Merino. However, the first result appeared when D. Lorenzini noticed in [13] that the graphs having critical group with one invariant factor equal to 1 consist only of the complete graphs. After, C. Merino in [16] posed interest on the characterization of for the cases with $2$ and $3$ invariant factors equal to 1. In this sense, few attempts have been done. For instance, in [18] it was characterized the graphs having critical group with 2 invariant factors equal to 1 whose third invariant factor is equal to $n$, $n-1$, $n-2$, or $n-3$. In [6] the characterizations of the same graphs but with a cut vertex, and with number of independent cycles equal to $n-2$ are given. Recently, a complete characterization of the graphs having critical group with two invariant factors equal to 1 was obtained in [2].

In this section, we will focus in giving a characterization of the digraphs whose critical group has one invariant factor equal to 1, and the characterization of the digraphs whose Smith group has one invariant factor equal to one.

Let us recall the concepts of adjacency matrix, Laplacian matrix, Smith group and critical group. The adjacency matrix $A(D)$ and Laplacian matrix $L(D)$ of $D$ are the evaluation of $-L(D,X_{D})$ and $L(D,X_{D})$ at $X={\bf 0}$ and at $X=\deg^{+}(D)$, respectively, where $\deg^{+}(D)$ is the out-degree vector of $D$. By considering a $m\times n$ matrix $M$ with integer entries as a linear map $M:\mathbb{Z}^{n}\rightarrow\mathbb{Z}^{m}$, the cokernel of $M$ is the quotient module $\mathbb{Z}^{n}/{\rm Im}\,M$. This finitely generated abelian group becomes a graph invariant when we take the matrix $M$ to be the adjacency or Laplacian matrix of the graph.

The cokernel of $A(D)$ is known as the Smith group of the digraph $D$ and is denoted $S(D)$, and the torsion part of the cokernel of $L(D)$ is knwon as the critical group $K(D)$ of $D$. For a survey on Smith groups see [19]. The critical group is especially interesting; for connected graphs since its order is equal to the number of spanning trees of the graph. The critical group has been studied intensively over the last 30 years on several contexts: the group of components [13, 14], the Picard group [4, 5], the Jacobian group [4, 5], the sandpile group [7], chip-firing game [5, 16], or Laplacian unimodular equivalence [10, 17].

One way to compute the cokernel of $M$ is by means of the Smith normal form and its associated invariant factors, see [12, Theorem 1.4]. Thus, the cokernel of $M$ can be described as: $coker(M)\cong\mathbb{Z}_{f_{1}}\oplus\mathbb{Z}_{f_{2}}\oplus\cdots\oplus\mathbb{Z}_{f_{r}}$, where $f_{1},f_{2},...,f_{r}$ are positive integers with $f_{i}\mid f_{j}$ for all $i\leq j$. These integers are called invariant factors of $M$. We might refer the reader to the Stanley’s survey [20] on the Smith normal forms in combinatorics for more details in the topic.

Computation of the Smith normal form of the adjacency or Laplacian matrix is a standard technique to determine the Smith or critical group of a graph. It is well known that this can be achieved through integral row and column operations. One more way (see [1, 9]) of determining the structure is by working directly within the particular abelian group to identify a cyclic decomposition for it. For it, we need to know the orders of the group, exhibit a set of elements and show that their orders divide the orders of the cyclic factors, and finally, show these elements do indeed generate the group.

An alternative in computing the invariant factors is by means of the following formula (see [11]). If $\Delta_{i}(M)$ is the greatest common divisor of the $i$-minors of the matrix $M$, then the $i$-th invariant factor $f_{i}$ is equal to $\Delta_{i}(M)/\Delta_{i-1}(M)$, where $\Delta_{0}(M)=1$. The following result uses this formula to build a bridge between the critical groups and critical ideals.

Proposition 4.1**.**

[8]** If $\deg^{+}(D)=(\deg^{+}_{D}(v_{1}),...,\deg^{+}_{D}(v_{n}))$ is the out–degree vector of $D$, and $f_{1}\mid\cdots\mid f_{n-1}$ are the invariant factors of $K(D)$, then

[TABLE]

Thus, if the critical ideal $I_{i}(D,X_{D})$ is trivial, then $\Delta_{i}(L(D))$ and $f_{i}$ are equal to $1$. Equivalently, if $\Delta_{i}(L(D))$ and $f_{i}$ are not equal to $1$, then the critical ideal $I_{i}(D,X_{D})$ is not trivial. The counterpart to the algebraic co-rank, in the critical group, is the number $f_{1}(D)$ of invariant factors equal to 1.

[TABLE]

Moreover, $\mathcal{G}_{i}\subseteq\Gamma_{\leq i}$ for all $i\geq 0$. By Proposition 4.1, to obtain the classification of the digraphs whose critical group has exactly one invariant factor equal to one, we only need to evaluate the out-degree of the vertices in the second critical ideal corresponding to each digraph in Lemma 3.2.

Corollary 4.2**.**

The critical group of a connected digraph has exactly one invariant factor equal to 1 if and only if is isomorphic to the digraph $\Lambda_{n_{1},n_{2},n_{3}}$ where $n_{1},n_{2},n_{3}$ satisfy one of the following conditions.

• •

$n_{1},n_{2},n_{3}\geq 1$,

• •

$n_{1},=n_{2}=1$, $n_{3}=0$,

• •

$n_{1},=n_{3}=1$, $n_{2}=0$,

• •

$n_{2},=n_{3}=1$, $n_{1}=0$,

• •

$n_{1}\geq 0$*, $n_{2}\geq 2$, $n_{3}\geq 0$, *

• •

$n_{1}=0$, $n_{2}=1$, $n_{3}\geq 2$,

• •

$n_{1}=1$, $n_{2}=0$, $n_{3}\geq 2$,

• •

$n_{1}\geq 2$, $n_{2}=0$, $n_{3}\geq 2$.

Proof.

We have that the out-degree of a vertex $v$ in $\Lambda_{n_{1},n_{2},n_{3}}$ is as follows.

[TABLE]

By evaluating each indeterminate at its associated out-degree in the second critical ideal of Lemma 3.2, we obtain that the only cases that the ideal trivializes are $n_{1}\geq 2,n_{2}=1,n_{3}=0$ and $n_{1}\geq 2,n_{2}=0,n_{3}=1$. ∎

Similar arguments used in [8] give us the following result for the Smith group.

Proposition 4.3**.**

If $f_{1}\mid\cdots\mid f_{n-1}$ are the invariant factors of $S(D)$, then

[TABLE]

By evaluating each indeterminate at zero in the second critical ideal of Lemma 3.2, we obtain the characterization.

Corollary 4.4**.**

The Smith group of a connected digraph has exactly one invariant factor equal to 1 if and only if is isomorphic to the digraph $\Lambda_{n_{1},n_{2},n_{3}}$ where $n_{1},n_{2},n_{3}$ satisfy one of the following conditions.

• •

$n_{1},=n_{2}=1$, $n_{3}=0$,

• •

$n_{1},=n_{3}=1$, $n_{2}=0$,

• •

$n_{2},=n_{3}=1$, $n_{1}=0$,

• •

$n_{1}\geq 2$, $n_{2}=1$, $n_{3}=0$,

• •

$n_{1}=0$, $n_{2}=1$, $n_{3}\geq 2$,

• •

$n_{1}=1$, $n_{2}=0$, $n_{3}\geq 2$,

• •

$n_{1}\geq 2$, $n_{2}=0$, $n_{3}=1$,

• •

$n_{1}\geq 2$, $n_{2}=0$, $n_{3}\geq 2$.

Appendix A SAGE code for verifying $\mathfrak{F}\subseteq{\mathbf{Forb}}(\Gamma_{\leq 1})$

⬇

Forbidden digraphs with 3 vertices

F3 = [];

F3.append(DiGraph({0:[2],2:[1]}, name=’F31’));

F3.append(DiGraph({0:[2],1:[2],2:[0]}, name=’F32’));

F3.append(DiGraph({0:[2],2:[0,1]}, name=’F33’));

F3.append(DiGraph({0:[2],1:[2],2:[0,1]}, name=’F34’));

F3.append(DiGraph({0:[1],1:[2],2:[0]}, name=’F35’));

F3.append(DiGraph({0:[1,2],1:[0],2:[1]}, name=’F36’));

F3.append(DiGraph({0:[1,2],1:[0,2],2:[0]}, name=’F37’));

Forbidden digraphs with 4 vertices

F4 = [];

F4.append(DiGraph({0:[2,3],1:[3]}, name=’F41’));

F4.append(DiGraph({0:[2,3],1:[3],2:[3]}, name=’F42’));

F4.append(DiGraph({0:[2],3:[0,1,2]}, name=’F43’));

F4.append(DiGraph({0:[2,3],1:[3],2:[0,3]}, name=’F44’));

F4.append(DiGraph({0:[2],1:[2],3:[0,1,2]}, name=’F45’));

F4.append(DiGraph({0:[2],2:[0],3:[0,1,2]}, name=’F46’));

F4.append(DiGraph({0:[1,2,3],1:[2,3],2:[3]}, name=’F47’));

F4.append(DiGraph({0:[1,2,3],1:[0,2,3],2:[3]}, name=’F48’));

F4.append(DiGraph({0:[1,2,3],1:[0,2,3],2:[3],3:[2]}, name=’F49’));

F4.append(DiGraph({0:[1],1:[0],2:[0,1,3],3:[0,1]}, name=’F410’));

def Gamma(L,n):

for i in range(n+1):

  I = R.ideal(L.minors(i))

  if( I.is_one() == True ):

     next

  else:

     return i-1

return n

def isForb(L,n):

for i in range(n):

  S = range(n)

  S.remove(i)

  I = R.ideal(L[S,S].minors(2))

  if( I.is_one() == True ):

     return "not forbidden"

  else:

     next

return "forbidden"

for F in F3:

n = len(F)

R = macaulay2.ring("ZZ","[x0,x1,x2]").to_sage();

R.inject_variables();

Laplacian = diagonal_matrix(list(R.gens())) - F.adjacency_matrix()

F.show()

print(’Forbidden graph ’ + F.name() + ’ has Gamma = ’ + str(Gamma(Laplacian,n)) + ’, and is ’ + isForb(Laplacian,n) + ’\n’)

for F in F4:

n = len(F)

R = macaulay2.ring("ZZ","[x0,x1,x2,x3]").to_sage();

R.inject_variables();

Laplacian = diagonal_matrix(list(R.gens())) - F.adjacency_matrix()

F.show()

print(’Forbidden graph ’ + F.name() + ’ has Gamma = ’ + str(Gamma(Laplacian,n)) + ’, and is ’ + isForb(Laplacian,n) + ’\n’)