Graphs of bounded depth-$2$ rank-brittleness

TL;DR

This paper characterizes certain graph classes closed under vertex-minors, using a tree-based measure called depth-2 rank-brittleness, and shows large values imply the presence of specific subgraphs as vertex-minors.

Contribution

It introduces the concept of depth-2 rank-brittleness and provides a characterization linking it to forbidden subgraphs and vertex-minors.

Findings

Graphs with large depth-2 rank-brittleness contain specific subgraphs as vertex-minors.

The characterization involves a tree of radius 2 with leaves labeled by graph vertices.

The measure of width is based on maximum cut-rank induced by splitting internal nodes.

Abstract

We characterize classes of graphs closed under taking vertex-minors and having no $P_n$ and no disjoint union of $n$ copies of the $1$-subdivision of $K_{1,n}$ for some $n$. Our characterization is described in terms of a tree of radius $2$ whose leaves are labelled by the vertices of a graph $G$, and the width is measured by the maximum possible cut-rank of a partition of $V(G)$ induced by splitting an internal node of the tree to make two components. The minimum width possible is called the depth-$2$ rank-brittleness of $G$. We prove that for all $n$, every graph with sufficiently large depth-$2$ rank-brittleness contains $P_n$ or disjoint union of $n$ copies of the $1$-subdivision of $K_{1,n}$ as a vertex-minor.

Equations

$${\mathrm{rbrit}}_1(G)\ge {\mathrm{rbrit}}_2(G)\ge {\mathrm{rbrit}}_3(G)\ge \cdots .$$

$$\{P_1,P_2,P_3,P_4,\dots \}⊈\mathcal{C}\text{ and }\{T_{2,1},2T_{2,2},3T_{2,3},4T_{2,4},\dots \}⊈\mathcal{C}.$$

$$\max\left\{\rho_{G}\bigl{(}\bigcup_{i\in I}X_{i}\bigr{)}:I\subseteq\{1,2,\ldots,m\}\right\}.$$

$${\rho }_G(A)={\rho }_G((A\setminus \{w\})\cup \{v\})={\rho }_{G-v}(A\setminus \{w\})\le k.$$

$$M:=\lceil (R(k-1;2)-1)(R(k-1;2)-2{)}^{2(k-1)-2}/(R(k-1;2)-3)\rceil .$$

$${\beta }_{M-1}^{\rho }(G)\ge N$$

$$f_2(n)$$

$$f_1(n)$$

$$\sigma (n)$$

$$H:=(4f_1(n))K_{2B(f_2(n))}.$$

$$|S_i|=\lceil |C_i|/2\rceil =B(f_2(n))$$

$$G_1=\{\begin{array}{ll}G\ast v_1\ast v_2\ast \cdots \ast v_n-v-X_n& \text{if n is even},\\ G\ast v_1\ast v_2\ast \cdots \ast v_{n-1}-v-X_n& \text{if n is odd.}\end{array}$$

$$(K_t\boxminus K_t{)}_A^s,(K_t\boxminus \overline{K_t}{)}_A^s,(K_t⊠K_t{)}_A^s,\text{ and }(K_t⊠\overline{K_t}{)}_A^s$$

$$n\ge N:=R_{prod}(8^{\left(\genfrac{}{}{0ex}{}{m}{2}\right)},m;1,1,\dots ,1;t,t,\dots ,t).$$

$$\{P_1,P_2,P_3,P_4,\dots \}⊈\mathcal{C},$$

$$\{T_{2,1},2T_{2,2},3T_{2,3},4T_{2,4},\dots \}⊈\mathcal{C}.$$

$$(K_{n+2}\boxminus K_{n+2}{)}_A^{n+2},(K_{n+2}\boxminus \overline{K_{n+2}}{)}_A^{n+2},(K_{n+2}⊠K_{n+2}{)}_A^{n+2},(K_{n+2}⊠\overline{K_{n+2}}{)}_A^{n+2}$$

Full text

1 Introduction

Tree-depth is a graph parameter in the theory of sparse graph classes, which measures how far a graph is from being a star, introduced by Nešetřil and Ossona de Mendez [19]. An equivalent concept has been introduced a few times under the names like the vertex ranking number and the minimum height elemination tree [3, 5, 25]. It is known that a graph has large tree-depth if and only if it has a long path, see [20, Section 6.2].

For some applications, it is desirable to say that complete graphs are also very similar to stars. However, complete graphs have unbounded tree-depth. To design a graph parameter similar to tree-depth but more suitable for dense graph classes, DeVos, Kwon, and Oum [6] introduced the rank-depth of a graph. Roughly speaking, the rank-depth of a graph $G$ is defined in terms of a decomposition, which is a tree whose leaves are labelled by the vertices of $G$. A decomposition has two qualities, one of which is the radius of the tree, and the other is the maximum width of internal nodes, measured by some connectivity function of $G$. The rank-depth of a graph $G$ is defined as the minimum integer $k$ such that $G$ admits a decomposition of radius at most $k$ and width at most $k$. The detailed definition of rank-depth will be reviewed in Section 2. In fact, there was an equivalent concept called the shrub-depth of classes of graphs, introduced by Ganian, Hliněný, Nešetřil, Obdržálek, Ossona de Mendez, and Ramadurai [11, 12]. The definition of shrub-depth uses logical terms similar to the definition of clique-width [4], while the definition of rank-depth uses a tree-like decomposition similar to that of rank-width [24]. DeVos, Kwon, and Oum [6] showed that a class of graphs has bounded rank-depth if and only if it has bounded shrub-depth.

Hliněný, Kwon, Obdržálek, and Ordyniak [14] proposed the following conjecture, which we state in terms of rank-depth. To state their conjecture, we first introduce vertex-minors. The local complementation at a vertex $v$ of a graph $G$ is an operation to obtain a new graph $G*v$ from $G$ by removing all edges $xy$ between two adjacent pairs $x$, $y$ of neighbors of $v$ and adding edges $xy$ for all non-adjacent pairs $x$, $y$ of neighbors of $v$. A graph $H$ is a vertex-minor of a graph $G$ if $H$ can be obtained from $G$ by a sequence of local complementations and vertex deletions. It is known that the rank-depth of a vertex-minor of $G$ is at most the rank-depth of $G$ and so it is natural to think of an obstruction for graphs of bounded rank-depth in terms of vertex-minors. The following conjecture states that paths are obstructions for having bounded rank-depth. This conjecture was verified for graphs of rank-width $1$ by Novotný [21, Theorem 6.3.2].

Conjecture 1.1** (Hliněný, Kwon, Obdržálek, and Ordyniak [14]).**

A class $\mathcal{C}$ of graphs has bounded rank-depth if and only if there exists an integer $t$ such that no graph $G\in\mathcal{C}$ contains a path of length $t$ as a vertex-minor.

As a step towards Conjecture 1.1, we define a new parameter called depth-$d$ rank-brittleness for an integer $d$ by restricting the radius of the tree in the decomposition to be at most $d$ in the definition of rank-depth. The depth-$d$ rank-brittleness of a graph $G$ is the minimum integer $k$ such that $G$ admits a decomposition of radius at most $d$ and width at most $k$. We denote this parameter by $\operatorname{rbrit}_{d}(G)$. By definition, the rank-depth of a graph $G$ is at most $\max\{d,\operatorname{rbrit}_{d}(G)\}$ for all $d\geq 1$ and

[TABLE]

In Section 6, we will show that a graph of rank-depth $k$ has linear rank-width at most $k^{2}$.

A class $\mathcal{C}$ of graphs is a vertex-minor ideal if for every graph $G\in\mathcal{C}$, $\mathcal{C}$ contains all graphs isomorphic to vertex-minors of $G$. For a graph $H$, we write $nH$ for the disjoint union of $n$ copies of $H$. It is straightforward to deduce the following proposition by using Ramsey-type results. To see this, one can use Theorem 2.3, Ramsey’s theorem, and Lemma 2.2. It can be also seen as a special case of a theorem due to Kwon and Oum [18, Theorem 1.4], which is stated in Theorem 4.2.

Proposition 1.2**.**

A vertex-minor ideal $\mathcal{C}$ has bounded depth-$1$ rank-brittleness if and only if $\{K_{2},2K_{2},3K_{2},\ldots\}\not\subseteq\mathcal{C}$.

In this paper, we characterize classes of graphs of bounded depth-$2$ rank-brittleness in terms of forbidden vertex-minors. Let $T_{2,n}$ be the $1$-subdivision of $K_{1,n}$, see Figure 1. Here is our main theorem.

Theorem 1.3**.**

A vertex-minor ideal $\mathcal{C}$ has bounded depth-$2$ rank-brittleness if and only if

[TABLE]

Since $T_{2,n}$ contains $P_{5}$ if $n\geq 2$, we obtain the following corollary, confirming a weaker statement of Conjecture 1.1.

Corollary 1.4**.**

For every positive integer $n$, graphs with no vertex-minors isomorphic to $nP_{5}$ have bounded depth-$2$ rank-brittleness, bounded rank-depth, and bounded linear rank-width.

We sketch the proof of Theorem 1.3. It is straightforward to show that $P_{n}$ and $nT_{2,n}$ have large depth-$2$ rank-brittleness. We mainly show that for every fixed $n$, if a graph $G$ has sufficiently large depth-$2$ rank-brittleness, then it has a vertex-minor isomorphic to $P_{n}$ or $nT_{2,n}$. A theorem of Kwon and Oum [18, Theorem 1.4] will imply that every graph of large depth-$2$ rank-brittleness has a vertex-minor isomorphic to $aK_{b}$ for large $a$ and $b$. By taking a graph locally equivalent to $G$, we may assume that $G$ has an induced subgraph isomorphic to $aK_{b}$.

In Section 3, we prove that if a graph $H$ contains $3$ pairwise twins, then one of them can be removed without decreasing the depth-$2$ rank-brittleness. Using that, each component $C$ of $aK_{b}$ can be partitioned into at least $b/2$ sets such that vertices in distinct sets are not twins. By the Ramsey-type result on bipartite graphs, we will extract a large (induced) matching or an anti-matching or a half graph between $C$ and the rest. We find this for each component of $aK_{b}$. Then using the sunflower lemma and Ramsey’s theorem, we will clean up all the structures and find a vertex-minor isomorphic to $nT_{2,n}$ or $P_{n}$. Section 4 is devoted to describe all the intermediate structures. The proof of Theorem 1.3 is given in Section 5. Section 6 shows an inequality between linear rank-width and rank-depth and presents a corollary of Theorem 1.3 for graphs with no vertex-minors isomorphic to $nP_{5}$.

2 Preliminaries

All graphs in this paper are simple and undirected. For a graph $G$, we denote by $V(G)$ and $E(G)$ the vertex set and the edge set of $G$, respectively. Let $G$ be a graph. For $S\subseteq V(G)$, we denote by $G[S]$ the subgraph of $G$ induced by $S$, and for two disjoint vertex subsets $S$ and $T$ of $G$, we denote by $G[S,T]$ the bipartite graph with bipartition $(S,T)$ such that for $a\in S$ and $b\in T$, $a,b$ are adjacent in $G[S,T]$ if and only if they are adjacent in $G$. For $v\in V(G)$, we denote by $G-v$ the graph obtained from $G$ by removing $v$ and all edges incident with $v$. For a set $X$ of vertices, we denote by $G-X$ the graph obtained from $G$ by deleting all vertices in $X$ and all edges incident with those vertices. For $v\in V(G)$, the set of neighbors of $v$ in $G$ is denoted by $N_{G}(v)$, and the degree of $v$ is the size of $N_{G}(v)$. We denote by $A(G)$ the adjacency matrix of $G$.

For two disjoint vertex subsets $A$ and $B$ of $G$, we say that $A$ is complete to $B$ if every vertex in $A$ is adjacent to all vertices in $B$. Similarly, $A$ is anti-complete to $B$, if every vertex in $A$ is non-adjacent to all vertices in $B$. A clique is a set of pairwise adjacent vertices and an independent set is a set of pairwise non-adjacent vertices.

Two vertices $v$ and $w$ in a graph $G$ are called twins if $N_{G}(v)\setminus\{v,w\}=N_{G}(w)\setminus\{v,w\}$. Note that a set of pairwise twins is either a clique or an independent set.

Let $K_{n}$ denote the complete graph on $n$ vertices, and let $K_{1,n}$ denote the star with $n$ leaves. Let $P_{n}$ denote the path on $n$ vertices. For a graph $G$, we denote by $\overline{G}$ the complement of $G$, that is, two vertices $v$ and $w$ in $G$ are adjacent if and only if they are not adjacent in $\overline{G}$.

We write $R(n;k)$ to denote the minimum number $N$ such that every coloring of the edges of $K_{N}$ into $k$ colors induces a monochromatic complete subgraph on $n$ vertices. The classical theorem of Ramsey implies that $R(n;k)$ exists.

We also use the sunflower lemma. Let $\mathcal{F}$ be a family of sets. A subset $\{M_{1},M_{2},\ldots,M_{p}\}$ of $\mathcal{F}$ is a sunflower with core $A$ (possibly an empty set) and $p$ petals if for all distinct $i,j\in\{1,2,\ldots,p\}$, $M_{i}\cap M_{j}=A$.

Theorem 2.1** (Sunflower Lemma [9, Erdős and Rado]).**

Let $k$ and $p$ be positive integers, and $\mathcal{F}$ be a family of sets each of cardinality $k$. If $\lvert\mathcal{F}\rvert>k!(p-1)^{k}$, then $\mathcal{F}$ contains a sunflower with $p$ petals.

2.1 Vertex-minors

For a vertex $v$ in a graph $G$, to perform local complementation at $v$, replace the subgraph of $G$ induced on $N_{G}(v)$ by its complement graph. We write $G*v$ to denote the graph obtained from $G$ by applying local complementation at $v$. Two graphs $G$ and $H$ are locally equivalent if $G$ can be obtained from $H$ by a sequence of local complementations. A graph $H$ is a vertex-minor of a graph $G$ if $H$ is an induced subgraph of a graph which is locally equivalent to $G$.

2.2 Rank-depth and rank-brittleness

The cut-rank function of a graph $G$, denoted by $\rho_{G}(S)$ for a subset $S$ of $V(G)$, is defined as the rank of an $S\times(V(G)\setminus S)$ [math]-$1$ matrix over the binary field whose $(a,b)$-entry for $a\in S$, $b\notin S$ is $1$ if $a$, $b$ are adjacent and [math] otherwise. The cut-rank function is invariant under the local complementation, see Oum [22]. The cut-rank function satisfies the submodular inequality, that is, for all $X,Y\subseteq V(G)$, $\rho_{G}(X)+\rho_{G}(Y)\geq\rho_{G}(X\cap Y)+\rho_{G}(X\cup Y)$. The $\rho_{G}$-width of a partition $\mathcal{P}=(X_{1},X_{2},\ldots,X_{m})$ of $V(G)$, for some $m$, is

[TABLE]

A decomposition of a graph $G$ is a pair $(T,\sigma)$ of a tree $T$ with at least one internal node and a bijection $\sigma$ from $V(G)$ to the set of leaves of $T$. The radius of a decomposition $(T,\sigma)$ is defined to be the radius of the tree $T$. For an internal node $v\in V(T)$, the components of the graph $T-v$ give rise to a partition $\mathcal{P}_{v}$ of $V(G)$ by $\sigma$ and the width of $v$ is defined to be the $\rho_{G}$-width of $\mathcal{P}_{v}$. The width of the decomposition $(T,\sigma)$ is the maximum width of an internal node of $T$. We say that a decomposition $(T,\sigma)$ is a $(k,r)$-decomposition of $G$ if the width is at most $k$ and the radius is at most $r$. The rank-depth of a graph $G$, denoted by $\operatorname{rd}(G)$, is the minimum integer $k$ such that $G$ admits a $(k,k)$-decomposition. If $\lvert V(G)\rvert<2$, then there exists no decomposition and rank-depth is defined to be [math]. Note that every tree in a decomposition has radius at least $1$ and therefore the rank-depth of a graph is at least $1$ if $\lvert V(G)\rvert\geq 2$.

The depth-$d$ rank-brittleness of a graph $G$, denoted by $\operatorname{rbrit}_{d}(G)$, is the minimum integer $k$ such that $G$ admits a $(k,d)$-decomposition. If $\lvert V(G)\rvert<2$, then we define $\operatorname{rbrit}_{d}(G)=0$. Note that the depth-$1$ rank-brittleness of a graph $G$ is equal to $\max_{A\subseteq V(G)}\rho_{G}(A)$.

2.3 Constructions of common graphs

For two graphs $G$ and $H$ on the disjoint vertex sets, each having $n$ vertices, we would like to introduce operations to construct graphs on $2n$ vertices by making the disjoint union of them and adding some edges between two graphs. Roughly speaking, $G\boxminus H$ will add a perfect matching, $G\boxtimes H$ will add the complement of a perfect matching, and $G\boxslash H$ will add a half graph. Formally, for two $n$-vertex graphs $G$ and $H$ with fixed ordering on the vertex sets $\{v_{1},v_{2},\ldots,v_{n}\}$ and $\{w_{1},w_{2},\ldots,w_{n}\}$ respectively, let $G\boxminus H$, $G\boxtimes H$, $G\boxslash H$ be graphs on the vertex set $V(G)\cup V(H)$ whose subgraph induced by $V(G)$ or $V(H)$ is $G$ or $H$, respectively such that for all $i,j\in\{1,2,\ldots,n\}$,

1. (i)

$v_{i}w_{j}\in E(G\boxminus H)$ if and only if $i=j$, 2. (ii)

$v_{i}w_{j}\in E(G\boxtimes H)$ if and only if $i\neq j$, 3. (iii)

$v_{i}w_{j}\in E(G\boxslash H)$ if and only if $i\geq j$.

See Figure 2 for illustrations of ${K_{5}}\boxminus\overline{K_{5}}$, ${K_{5}}\boxtimes\overline{K_{5}}$, and ${K_{5}}\boxslash\overline{K_{5}}$.

We will use the following lemma. Similar lemmas appeared in [17, Lemma 2.8], [14, Proposition 6.2], and [16, Lemma 5.6].

Lemma 2.2** (Kwon and Oum [18, Lemma 6.5]).**

Let $n$ be a positive integer.

1. (1)

$\overline{K_{n}}\boxslash\overline{K_{n}}$* is locally equivalent to $P_{2n}$.* 2. (2)

${K_{n}}\boxslash\overline{K_{n}}$* is locally equivalent to $P_{2n}$.* 3. (3)

If $n\geq 2$, then ${K_{n}}\boxslash{K_{n}}$ has a vertex-minor isomorphic to $P_{2n-2}$.

We also use the Ramsey-type result on bipartite graphs without twins.

Theorem 2.3** **(Ding, Oporowski, Oxley, and

Vertigan [8]).

For every positive integer $n$, there exists an integer $B(n)$ such that for every bipartite graph $G$ with a bipartition $(S,T)$, if no two vertices in $S$ have the same set of neighbors and $\lvert S\rvert\geq B(n)$, then $S$ and $T$ have $n$-element subsets $S^{\prime}$ and $T^{\prime}$, respectively, such that $G[S^{\prime},T^{\prime}]$ is isomorphic to $\overline{K_{n}}\boxminus\overline{K_{n}}$, $\overline{K_{n}}\boxslash\overline{K_{n}}$, or $\overline{K_{n}}\boxtimes\overline{K_{n}}$.

For a positive integer $t$, a $2\times 2$ matrix $A=(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix})$, and a binary operator $\odot\in\{\boxminus,\boxtimes,\boxslash\}$ on two graphs of the same number of vertices, we define $(G\odot H)^{t}_{A}$ as the graph on the disjoint union of $t$ copies of $G\odot H$ such that for all $1\leq i<j\leq t$,

1. (i)

the $i$-th copy of $G$ is complete to the $j$-th copy of $G$ if $a=1$ and anti-complete if $a=0$, 2. (ii)

the $i$-th copy of $G$ is complete to the $j$-th copy of $H$ if $b=1$, and anti-complete if $b=0$, 3. (iii)

the $i$-th copy of $H$ is complete to the $j$-th copy of $G$ if $c=1$, and anti-complete if $c=0$, 4. (iv)

the $i$-th copy of $H$ is complete to the $j$-th copy of $H$ if $d=1$, and anti-complete if $d=0$.

See Figure 3 for an illustration.

3 Lemma on three twins

In this section, we prove that if a graph has three pairwise twins, then one of them can be removed without decreasing its depth-$d$ rank-brittleness for $d\geq 2$. It holds for all $d\geq 2$ but we will only use it for $d=2$ later.

Lemma 3.1**.**

Let $d\geq 2$ be an integer. Let $v$, $w$, $z$ be vertices of a graph $G$ that are pairwise twins. Then $\operatorname{rbrit}_{d}(G)=\operatorname{rbrit}_{d}(G-v)$.

Proof.

The inequality $\operatorname{rbrit}_{d}(G)\geq\operatorname{rbrit}_{d}(G-v)$ is trivial by definition. We will show that if $G-v$ has a $(k,d)$-decomposition $(T,\sigma)$, then $G$ also has a $(k,d)$-decomposition.

Let $r$ be a node of $T$, called a root of $T$, which has distance at most $d$ to every node of $T$. We may assume that $r$ is not a leaf node. Let $a$ be the leaf of $T$ with $a=\sigma(w)$ and $b$ be the parent of $a$ in $T$, which is the unique neighbor of $a$ in $T$. We obtain a decomposition $(T_{1},\sigma_{1})$ of $G$ as follows: $T_{1}$ is the tree obtained from $T$ by adding a new node $a^{\prime}$ adjacent to $b$, and assign $\sigma_{1}(v):=a^{\prime}$ and $\sigma_{1}(x):=\sigma(x)$ for all $x\in V(G)\setminus\{v\}$. We claim that $(T_{1},\sigma_{1})$ is a $(k,d)$-decomposition of $G$. Clearly, $T_{1}$ has radius at most $d$. So, it is sufficient to show that every internal node of $T_{1}$ has width at most $k$. For each internal node $t$ of $T_{1}$, let $\mathcal{P}_{t}$ be the partition of $V(G)$ derived from the components of $T_{1}-t$ by $\sigma_{1}^{-1}$.

For an internal node $t\neq b$ in $T_{1}$, the width of $t$ in $(T_{1},\sigma_{1})$ is the same as its width in the decomposition $(T,\sigma)$ because $v$ and $w$ are twins of $G$ and $v$ and $w$ lie on the same part of $\mathcal{P}_{t}$.

We claim that the width of $b$ in $(T_{1},\sigma_{1})$ is at most $k$. Let $\mathcal{P^{\prime}}\subseteq\mathcal{P}_{b}$ and $A:=\bigcup_{X\in\mathcal{P^{\prime}}}X$. In the bipartition $(A,V(G)\setminus A)$, if $v$ is contained in a part together with $w$ or $z$, then the bipartition obtained by removing $v$ arises in the decomposition $(T,\sigma)$ as well. So, without loss of generality, we may assume that $w,z\in A$ and $v\in V(G)\setminus A$. But in this case, as $v,w,z$ are pairwise twins, the bipartition obtained by exchanging $v$ and $w$ has the same cut-rank. As $w$ is a single-vertex part of $\mathcal{P}_{b}$, the bipartition $(A\setminus\{w\},V(G)\setminus(A\setminus\{w\}))$ arises in the decomposition $(T,\sigma)$. So,

[TABLE]

We conclude that the width of every internal node of $T_{1}$ is at most $k$. ∎

4 Reducing to two cases

We recall the definition of rank $k$-brittleness [18]. The rank $k$-brittleness of a graph $G$, denoted by $\beta^{\rho}_{k}(G)$, is the minimum $\rho_{G}$-width of all partitions of $V(G)$ into parts of size at most $k$.

Lemma 4.1**.**

$\operatorname{rbrit}_{2}(G)\leq\max(2,k,\beta^{\rho}_{k}(G))$* for every positive integer $k$.*

Proof.

Suppose that $(X_{1},X_{2},\ldots,X_{m})$ is a partition of $V(G)$ whose $\rho_{G}$-width is $\beta^{\rho}_{k}(G)$. We create a decomposition $(T,\sigma)$ of $G$ as follows. Let $r$ be the root of $T$, and let $r_{1},r_{2},\ldots,r_{m}$ be the children of $T$, and each $r_{i}$ has exactly $\lvert X_{i}\rvert$ leaves adjacent to $r_{i}$, and we assign $X_{i}$ to these leaves by $\sigma$. It is easy to see that each $r_{i}$ has width at most $k$, and the root $r$ has width at most $\beta^{\rho}_{k}(G)$. Thus, $\operatorname{rbrit}_{2}(G)\leq\max(2,k,\beta^{\rho}_{k}(G))$. ∎

Kwon and Oum [18] proved the following.

Theorem 4.2** (Kwon and Oum [18, Theorem 1.4]).**

For every positive integer $n$, there exists $N$ such that every graph $G$ with $\beta^{\rho}_{k}(G)\geq N$ contains a vertex-minor isomorphic to $nH$ for some connected graph $H$ on $k+1$ vertices.

Every large connected graph has a long induced path or a vertex of large degree.

Proposition 4.3** (See Diestel [7, Proposition 1.3.3]).**

For integers $k>3$ and $\ell>0$, every connected graph on at least $(k-1)(k-2)^{\ell-2}/(k-3)$ vertices contains a vertex of degree at least $k$ or an induced path on $\ell$ vertices.

As a corollary we deduce the following. Essentially its proof is almost identical to the proof of [18, Theorem 1.6].

Corollary 4.4**.**

For all positive integers $k$ and $n$, there exists $N=h(n,k)$ such that every graph $G$ with depth-$2$ rank-brittleness at least $N$ has a vertex-minor isomorphic to $nK_{k}$.

Proof.

We may assume that $k>3$ by increasing $k$ if necessary. Let

[TABLE]

By Proposition 4.3, every connected graph with at least $M$ vertices has a vertex of degree at least $R(k-1;2)$ or an induced subgraph isomorphic to $P_{2(k-1)}$. By Theorem 4.2, there exists $N$ such that $N\geq M$ and every graph $G$ with $\beta^{\rho}_{M-1}(G)\geq N$ contains $nH$ for some connected graph $H$ on $M$ vertices. By Lemma 2.2(2), if $H$ contains $P_{2k}$ as an induced subgraph, then $H$ contains $K_{k}$ as a vertex-minor. If $H$ contains a vertex of degree at least $R(k-1;2)$, then it contains $K_{1,k-1}$ or $K_{k}$ as an induced subgraph. In all cases, $H$ contains $K_{k}$ as a vertex-minor and so $nH$ contains $nK_{k}$ as a vertex-minor.

If $G$ has depth-$2$ rank-brittleness at least $N$, then by Lemma 4.1,

[TABLE]

and therefore $G$ has a vertex-minor isomorphic to $nK_{k}$. ∎

Proposition 4.5**.**

For every integer $n\geq 2$, there exists an integer $\sigma(n)$ such that every graph $G$ of depth-$2$ rank-brittleness at least $\sigma(n)$ contains a vertex-minor $G^{\prime}$ satisfying one of the following.

1. (i)

$V(G^{\prime})=X_{1}^{*}\cup X_{2}^{*}\cup\cdots\cup X_{n}^{*}\cup Q^{*}$* for disjoint sets $X_{1}^{*}$, $X_{2}^{*},\ldots,X_{n}^{*},Q^{*}$ of $n$ vertices such that each $X_{i}^{*}$ is a clique in $G^{\prime}$, $G^{\prime}[X_{1}^{*}\cup X_{2}^{*}\cup\cdots\cup X_{n}^{*}]$ is isomorphic to $nK_{n}$, and either $K_{n}\boxminus\overline{K_{n}}$ or $K_{n}\boxtimes\overline{K_{n}}$ is isomorphic to all $G^{\prime}[X_{i}^{*}\cup Q^{*}]$.* 2. (ii)

$V(G^{\prime})=X_{1}^{*}\cup X_{2}^{*}\cup\cdots\cup X_{n}^{*}\cup Y_{1}^{*}\cup Y_{2}^{*}\cup\cdots\cup Y_{n}^{*}$* for disjoint sets $X_{1}^{*}$, $X_{2}^{*},\ldots,X_{n}^{*},Y_{1}^{*},Y_{2}^{*},\ldots,Y_{n}^{*}$ of $n$ vertices such that each $X_{i}^{*}$ is a clique in $G^{\prime}$, $G^{\prime}[X_{1}^{*}\cup X_{2}^{*}\cup\cdots\cup X_{n}^{*}]$ is isomorphic to $nK_{n}$, and one of $K_{n}\boxminus\overline{K_{n}}$, $K_{n}\boxminus K_{n}$, $K_{n}\boxtimes\overline{K_{n}}$, and $K_{n}\boxtimes K_{n}$ is isomorphic to all $G^{\prime}[X_{i}^{*}\cup Y_{i}^{*}]$.* 3. (iii)

$G^{\prime}$* is isomorphic to $P_{n}$.*

Proof.

If $G$ contains a component with at least $3$ vertices, then it contains a vertex-minor isomorphic to $P_{3}$. Thus, we may assume that $n\geq 4$.

Let $B$ be the function defined in Theorem 2.3, and $h$ be the function defined in Corollary 4.4. Let

[TABLE]

We may assume that no proper vertex-minor of $G$ has depth-$2$ rank-brittleness at least $\sigma(n)$. By Lemma 3.1, every graph locally equivalent to $G$ has no three vertices that are pairwise twins.

By Corollary 4.4, $G$ has a vertex-minor isomorphic to

[TABLE]

We may assume that $(4f_{1}(n))K_{2B(f_{2}(n))}$ is an induced subgraph of $G$ by applying local complementations. Let $C_{1},C_{2},\ldots,C_{4f_{1}(n)}$ be the set of connected components of $H$, and let $U:=V(G)\setminus V(H)$.

Observe that $G$ has no three vertices that are pairwise twins. It means that in each $C_{i}$, there are no three vertices that have the same neighborhood on $U$ in $G$, and thus each $C_{i}$ contains a subset $S_{i}$ with

[TABLE]

that have pairwise distinct sets of neighbors on $U$.

Now, we consider the bipartite graph $G[S_{i},U]$ for each $i$. In this bipartite graph, since vertices in $S_{i}$ have distinct neighborhoods on $U$ and $\lvert S_{i}\rvert=B(f_{2}(n))$, by Theorem 2.3, there exist $X_{i}^{\prime}\subseteq S_{i}$ and $Y_{i}^{\prime}\subseteq U$ such that $G[X_{i}^{\prime},Y_{i}^{\prime}]$ is isomorphic to $\overline{K_{f_{2}(n)}}\boxminus\overline{K_{f_{2}(n)}}$, $\overline{K_{f_{2}(n)}}\boxslash\overline{K_{f_{2}(n)}}$, or $\overline{K_{f_{2}(n)}}\boxtimes\overline{K_{f_{2}(n)}}$ for each $i\in\{1,2,\ldots,4f_{1}(n)\}$.

As $f_{2}(n)=R(2n;2)$, by Ramsey’s theorem, there exist $X_{i}\subseteq X_{i}^{\prime}$ and $Y_{i}\subseteq Y_{i}^{\prime}$ with $\lvert X_{i}\rvert=\lvert Y_{i}\rvert=2n$ where

• •

$Y_{i}$ is a clique or an independent set in $G$,

• •

$G[X_{i},Y_{i}]$ is isomorphic to $\overline{K_{2n}}\boxminus\overline{K_{2n}}$, $\overline{K_{2n}}\boxslash\overline{K_{2n}}$, or $\overline{K_{2n}}\boxtimes\overline{K_{2n}}$.

This can be done by selecting $Y_{i}$ from $Y_{i}^{\prime}$ by using Ramsey’s theorem and then selecting $X_{i}$ by using the relation between $X_{i}^{\prime}$ and $Y_{i}^{\prime}$. If $G[X_{i},Y_{i}]$ is isomorphic to $\overline{K_{2n}}\boxslash\overline{K_{2n}}$ for some $i$, then $G$ contains a vertex-minor isomorphic to a path on $4n-2\geq n$ vertices by Lemma 2.2. Thus, we may assume that $G[X_{i},Y_{i}]$ is isomorphic to $\overline{K_{2n}}\boxminus\overline{K_{2n}}$ or $\overline{K_{2n}}\boxtimes\overline{K_{2n}}$ for all $i\in\{1,2,\ldots,4f_{1}(n)\}$. So $G[X_{i}\cup Y_{i}]$ for each $i$ is isomorphic to ${K_{2n}}\boxminus\overline{K_{2n}}$, ${K_{2n}}\boxtimes\overline{K_{2n}}$, ${K_{2n}}\boxminus{K_{2n}}$, or ${K_{2n}}\boxtimes{K_{2n}}$. By the pigeonhole principle, we may assume that for all $i\in\{1,2,\ldots,f_{1}(n)\}$, all graphs $G[X_{i}\cup Y_{i}]$ are isomorphic to exactly one of ${K_{2n}}\boxminus\overline{K_{2n}}$, ${K_{2n}}\boxtimes\overline{K_{2n}}$, ${K_{2n}}\boxminus{K_{2n}}$, and ${K_{2n}}\boxtimes{K_{2n}}$.

We now apply Theorem 2.1, the sunflower lemma, to sets $Y_{1},Y_{2},\ldots,Y_{f_{1}(n)}$. As we choose $f_{1}(n)>(2n)!n^{2n}$, $\{Y_{1},Y_{2},\ldots,Y_{f_{1}(n)}\}$ contains a sunflower $\mathcal{F}$ with $n+1$ petals. We may assume that $\mathcal{F}=\{Y_{1},Y_{2},\ldots,Y_{n+1}\}$. Let $Q$ be the core of $\mathcal{F}$, that is $\bigcap_{i=1}^{n+1}Y_{i}$. Note that either $Q$ has at least $n+1$ vertices, or $Y_{i}\setminus Q$ has at least $n$ vertices for all $i=1,2,\ldots,n+1$. We divide into two cases depending on the size of the core.

First, suppose that $\lvert Q\rvert\geq n+1$. Let $Q^{*}$ be a subset of $Q$ with $\lvert Q^{*}\rvert=n$. For $i=1,2,\ldots,n$, let $X_{i}^{*}$ be the set of vertices in $X_{i}$ paired with vertices in $Q^{*}\subseteq Y_{i}$ in the graph $G[X_{i}\cup Y_{i}]$. Then each $X_{i}^{*}$ is a clique and $G[X_{1}^{*}\cup X_{2}^{*}\cup\cdots\cup X_{n}^{*}]$ is isomorphic to $nK_{n}$. Let $w\in Q\setminus Q^{*}$.

If all $G[X_{i}^{*}\cup Q^{*}]$ are isomorphic to $K_{n}\boxtimes K_{n}$, then $X_{n+1}$ has a vertex $v$ adjacent to all vertices in $Q^{*}$. In this case, we take $G^{\prime}:=G*v$. Then for every $i\in\{1,2,\ldots,n\}$, $G^{\prime}[X_{i}^{*}\cup Q^{*}]$ is isomorphic to $K_{n}\boxtimes\overline{K_{n}}$.

If all $G[X_{i}^{*}\cup Q^{*}]$ are isomorphic to $K_{n}\boxminus K_{n}$, then we take $G^{\prime}:=G*w$. Then all $G^{\prime}[X_{i}^{*}\cup Q^{*}]$ are isomorphic to $K_{n}\boxminus\overline{K_{n}}$.

If all $G[X_{i}^{*}\cup Q^{*}]$ are isomorphic to $K_{n}\boxminus\overline{K_{n}}$ or $K_{n}\boxtimes\overline{K_{n}}$, then we take $G^{\prime}:=G$.

We conclude that $G$ has a vertex-minor $G^{\prime}$ on $X_{1}^{*}\cup X_{2}^{*}\cup\cdots\cup X_{n}^{*}\cup Q^{*}$ such that each $X_{i}^{*}$ is a clique in $G^{\prime}$, $X_{i}^{*}$ is anti-complete to $X_{j}^{*}$ for all $i\neq j$ in $G^{\prime}$, and one of $K_{n}\boxminus\overline{K_{n}}$ or $K_{n}\boxtimes\overline{K_{n}}$ is isomorphic to $G^{\prime}[X_{i}^{*}\cup Q^{*}]$ for all $i\in\{1,2,\ldots,n\}$. So, $G^{\prime}$ provides a desired vertex-minor of the first type.

Now it remains to consider the case that $\lvert Q\rvert\leq n$. Then for all $i\in\{1,2,\ldots,n\}$, $\lvert Y_{i}\setminus Q\rvert\geq n$. For each $i=1,2,\ldots,n$, let $Y_{i}^{*}$ be a subset of $Y_{i}\setminus Q$ with $\lvert Y_{i}^{*}\rvert=n$. For $i=1,2,\ldots,n$, let $X_{i}^{*}$ be the set of vertices in $X_{i}$ paired with vertices in $Y_{i}^{*}$ in the graph $G[X_{i}\cup Y_{i}]$. Then we deduce that $G[X_{1}^{*}\cup X_{2}^{*}\cup\cdots\cup X_{n}^{*}]$ is isomorphic to $nK_{n}$ and for all $i=1,2,\ldots,n$, all $G[X_{i}^{*}\cup Y_{i}^{*}]$ are isomorphic to exactly one of $K_{n}\boxminus\overline{K_{n}}$, $K_{n}\boxtimes\overline{K_{n}}$, $K_{n}\boxminus K_{n}$, or $K_{n}\boxtimes K_{n}$. and $X_{1}^{*},X_{2}^{*},\ldots,X_{n}^{*},Y_{1}^{*},Y_{2}^{*},\ldots,Y_{n}^{*}$ are disjoint. So, $(X_{1}^{*},X_{2}^{*},\ldots,X_{n}^{*})$ and $(Y_{1}^{*},Y_{2}^{*},\ldots,Y_{n}^{*})$ provide a desired induced subgraph of the second type. ∎

In the rest, we will find a vertex-minor isomorphic to $P_{n}$ or $nT_{2,n}$ when a given graph satisfies (i) or (ii) of Proposition 4.5.

4.1 The first case

Lemma 4.6**.**

Let $n\geq 2$ be an integer. Let $G$ be a graph on $n^{2}+n$ vertices such that $V(G)=X_{1}\cup X_{2}\cup\cdots\cup X_{n}\cup Q$ for disjoint sets $X_{1}$, $X_{2},\ldots,X_{n},Q$ of $n$ vertices, each $X_{i}$ is a clique in $G$, $G[X_{1}\cup X_{2}\cup\cdots\cup X_{n}]$ is isomorphic to $nK_{n}$, and all $G[X_{i}\cup Q]$ are isomorphic to $K_{n}\boxminus\overline{K_{n}}$. Then $G$ has a vertex-minor isomorphic to $P_{3n-1}$.

Proof.

Let $v_{1}$, $v_{2}$, $\ldots$, $v_{n}$ be an arbitrary enumeration of vertices in $Q$. For each $i\in\{1,2,\ldots,n-1\}$, there are two vertices $x_{i}$, $y_{i}$ in $X_{i}$ such that $N_{G}(x_{i})\cap Q=\{v_{i}\}$, $N_{G}(y_{i})\cap Q=\{v_{i+1}\}$. Let $x_{n}$ be the neighbor of $v_{n}$ in $X_{n}$. Then $v_{1}x_{1}y_{1}v_{2}x_{2}y_{2}v_{3}\cdots v_{n}x_{n}$ is an induced path on $3n-1$ vertices. ∎

Lemma 4.7**.**

Let $n\geq 2$ be an integer. Let $G$ be a graph on $n^{2}+n$ vertices such that $V(G)=X_{1}\cup X_{2}\cup\cdots\cup X_{n}\cup Q$ for disjoint sets $X_{1}$, $X_{2},\ldots,X_{n},Q$ of $n$ vertices, each $X_{i}$ is a clique in $G$, $G[X_{1}\cup X_{2}\cup\cdots\cup X_{n}]$ is isomorphic to $nK_{n}$, and all $G[X_{i}\cup Q]$ are isomorphic to $K_{n}\boxtimes\overline{K_{n}}$. Then $G$ has a vertex-minor isomorphic to $P_{4n-5}$.

Proof.

Let $v$ be a vertex in $Q$. Let $v_{i}$ be the vertex in $X_{i}$ non-adjacent to $v$. Let

[TABLE]

Observe that every vertex in $X_{i}\setminus\{v_{i}\}$ has degree $2$ in $G_{1}$ and $N_{G_{1}}(v_{i})=(X_{i}\setminus\{v_{i}\})\cup(Q\setminus\{v\})$ for all $i\in\{1,2,\ldots,n-1\}$. See Figure 4 for an illustration. Let $G_{2}$ be the graph obtained from $G_{1}$ by applying local complementations at all vertices in $\bigcup_{i=1}^{n-1}(X_{i}\setminus\{v_{i}\})$. It is easy to see that $G_{2}$ is obtained from $G_{1}$ by deleting all edges from $v_{i}$ to $Q\setminus\{v\}$ for all $i\leq n-1$. Then $N_{G_{2}}(v_{i})=X_{i}\setminus\{v_{i}\}$ and $G_{2}[(X_{i}\setminus\{v_{i}\})\cup(Q\setminus\{v\})]$ is isomorphic to $\overline{K_{n-1}}\boxminus\overline{K_{n-1}}$. Let $w_{1}$, $w_{2}$, $\ldots$, $w_{n-1}$ be an arbitrary enumeration of vertices in $Q\setminus\{v\}$. For each $i\in\{1,2,\ldots,n-2\}$, there are vertices $x_{i},y_{i}\in X_{i}\setminus\{v_{i}\}$ such that $N_{G_{2}}(x_{i})\cap Q=\{w_{i}\}$ and $N_{G_{2}}(y_{i})\cap Q=\{w_{i+1}\}$. Let $x_{n-1}$ be the neighbor of $w_{n-1}$ in $X_{n-1}$. Then $w_{i}x_{i}v_{i}y_{i}w_{i+1}$ is an induced path and so $w_{1}x_{1}v_{1}y_{1}w_{2}x_{2}v_{2}y_{2}\cdots w_{n-2}x_{n-2}v_{n-2}y_{n-2}w_{n-1}x_{n-1}v_{n-1}$ is an induced path on $n-1+3(n-2)+2=4n-5$ vertices in $G_{2}$. Thus, $G$ has a vertex-minor isomorphic to $P_{4n-5}$. ∎

4.2 The second case

We will use the product Ramsey theorem described below.

Theorem 4.8** ([26, Theorem 11.5]; See also [13]).**

Let $r,t$ be positive integers, and let $k_{1},k_{2},\ldots,k_{t}$ be nonnegative integers, and let $m_{1},m_{2},\ldots,m_{t}$ be integers with $m_{i}\geq k_{i}$ for each $i\in\{1,2,\ldots,t\}$. Then there exists an integer $R=R_{prod}(r,t;k_{1},k_{2},\ldots,k_{t};m_{1},m_{2},\ldots,m_{t})$ such that if $X_{1},X_{2},\ldots,X_{t}$ are sets with $\lvert X_{i}\rvert\geq R$ for each $i\in\{1,2,\ldots,t\}$, then for every function $f:{X_{1}\choose k_{1}}\times{X_{2}\choose k_{2}}\times\cdots\times{X_{t}\choose k_{t}}\rightarrow\{1,2,\ldots,r\}$, there exist an element $\alpha\in\{1,2,\ldots,r\}$ and subsets $Y_{1},Y_{2},\ldots,Y_{t}$ of $X_{1},X_{2},\ldots,X_{t}$, respectively, so that $\lvert Y_{i}\rvert\geq m_{i}$ for each $i\in\{1,2,\ldots,t\}$, and $f$ maps every element of ${Y_{1}\choose k_{1}}\times{Y_{2}\choose k_{2}}\times\cdots\times{Y_{t}\choose k_{t}}$ to $\alpha$.

Lemma 4.9**.**

For integers $s$ and $t$, there exist $M=f(s,t)$ and $N=g(s,t)$ such that for $m\geq M$ and $n\geq N$, if a graph $G$ has $2m$ disjoint $n$-vertex sets $X_{1}$, $X_{2},\ldots,X_{m},Y_{1},Y_{2},\ldots,Y_{m}$, each $X_{i}$ is a clique of $G$, $G[X_{1}\cup X_{2}\cup\cdots\cup X_{m}]$ is isomorphic to $mK_{n}$, and one of $K_{n}\boxminus\overline{K_{n}}$, $K_{n}\boxminus K_{n}$, $K_{n}\boxtimes\overline{K_{n}}$, and $K_{n}\boxtimes K_{n}$ is isomorphic to all $G[X_{i}\cup Y_{i}]$, then there exist indices $1\leq i_{1}<i_{2}<\cdots<i_{s}\leq m$ and subsets $X_{1}^{*}$, $X_{2}^{*}$, $\ldots$, $X_{s}^{*}$ of $X_{i_{1}}$, $X_{i_{2}}$, $\ldots$, $X_{i_{s}}$ respectively and subsets $Y_{1}^{*}$, $Y_{2}^{*}$, $\ldots$, $Y_{s}^{*}$ of $Y_{i_{1}}$, $Y_{i_{2}}$, $\ldots$, $\ldots$, $Y_{i_{s}}$ respectively such that the following hold.

1. (i)

$\lvert X_{i}^{*}\rvert=\lvert Y_{i}^{*}\rvert=t$* for all $1\leq i\leq s$,* 2. (ii)

one of $K_{t}\boxminus\overline{K_{t}}$, $K_{t}\boxminus K_{t}$, $K_{t}\boxtimes\overline{K_{t}}$, and $K_{t}\boxtimes K_{t}$ is isomorphic to all $G[X_{i}^{*}\cup Y_{i}^{*}]$ for all $1\leq i\leq s$, 3. (iii)

$X_{i}^{*}$* is complete to $Y_{j}^{*}$ for all $i<j$ or $X_{i}^{*}$ is anti-complete to $Y_{j}^{*}$ for all $i<j$,* 4. (iv)

$Y_{i}^{*}$* is complete to $X_{j}^{*}$ for all $i<j$ or $Y_{i}^{*}$ is anti-complete to $X_{j}^{*}$ for all $i<j$,* 5. (v)

$Y_{i}^{*}$* is complete to $Y_{j}^{*}$ for all $i<j$ or $Y_{i}^{*}$ is anti-complete to $Y_{j}^{*}$ for all $i<j$.*

In other words, $G$ has an induced subgraph isomorphic to one of

[TABLE]

for some [math]-$1$ matrix $A=(\begin{smallmatrix}0&b\\ c&d\end{smallmatrix})$.

Proof.

Let $m\geq M:=R(s;8)$ and let

[TABLE]

The first step of the proof is to clean up edges between $X_{i}\cup Y_{i}$ and $X_{j}\cup Y_{j}$ for distinct $i$ and $j$. We consider a function that maps $(v_{1},v_{2},\ldots,v_{m})$ for $v_{i}\in Y_{i}$ to an edge-coloring of $K_{m}$ with colors on the edges $ij$ based on the three possible adjacencies between a pair of $v_{i}$ and its unique neighbor or non-neighbor in $X_{i}$ and a pair of $v_{j}$ and its unique neighbor or non-neighbor in $X_{j}$. Each edge of $K_{m}$ will receive one of $2^{3}$ colors and the range of this function has at most $8^{\binom{m}{2}}$ edge-colorings of $K_{m}$. By Theorem 4.8, there exist subsets $X_{1}^{\prime}$, $X_{2}^{\prime}$, $\ldots$, $X_{m}^{\prime}$, $Y_{1}^{\prime}$, $Y_{2}^{\prime}$, $\ldots$, $Y_{m}^{\prime}$ of $X_{1}$, $X_{2},\ldots,X_{m},Y_{1},Y_{2},\ldots,Y_{m}$, respectively, such that

1. (i)

$\lvert X_{i}^{\prime}\rvert=\lvert Y_{i}^{\prime}\rvert=t$, 2. (ii)

for each $i\neq j$, $X_{i}^{\prime}$ is complete or anti-complete to $Y_{j}^{\prime}$, and $Y_{i}^{\prime}$ is complete or anti-complete to $Y_{j}^{\prime}$, 3. (iii)

one of $K_{t}\boxminus\overline{K_{t}}$, $K_{t}\boxminus K_{t}$, $K_{t}\boxtimes\overline{K_{t}}$, and $K_{t}\boxtimes K_{t}$ is isomorphic to all $G[X_{i}^{\prime}\cup Y_{i}^{\prime}]$.

Now our next goal is to take a subset of $\{1,2,\ldots,m\}$ by using Ramsey’s theorem. Let us color the edges $ij$ of $K_{m}$ ($i<j$) by the one of $8$ colors determined by the following:

• •

$X_{i}^{\prime}$ is complete to $Y_{j}^{\prime}$ or not.

• •

$Y_{i}^{\prime}$ is complete to $X_{j}^{\prime}$ or not.

• •

$Y_{i}^{\prime}$ is complete to $Y_{j}^{\prime}$ or not.

Then by Ramsey’s theorem, there exists a subset $I$ of $\{1,2,\ldots,m\}$ with $\lvert I\rvert=s$ such that every edge of $K_{m}$ has the same color. Let $I=\{i_{1},i_{2},\ldots,i_{s}\}$ for $i_{1}<i_{2}<\cdots<i_{s}$ and $X_{j}^{*}=X_{i_{j}}$, $Y_{j}^{*}=Y_{i_{j}}$ for $1\leq j\leq s$. This provides our conclusion. ∎

Now we will see that in many cases, we will have a vertex-minor isomorphic to $nT_{2,n}$.

Lemma 4.10**.**

Let $n$ be a positive integer.

1. (1)

${K_{n+1}}\boxminus\overline{K_{n+1}}$* contains a vertex-minor isomorphic to $T_{2,n}$.* 2. (2)

${K_{n+2}}\boxminus{K_{n+2}}$* contains a vertex-minor isomorphic to $T_{2,n}$.* 3. (3)

${K_{n+2}}\boxtimes\overline{K_{n+2}}$* contains a vertex-minor isomorphic to $T_{2,n}$.* 4. (4)

${K_{n+1}}\boxtimes{K_{n+1}}$* contains a vertex-minor isomorphic to $T_{2,n}$.*

Therefore, if $A=(\begin{smallmatrix}0&0\\ 0&0\end{smallmatrix})$, then all of $({K_{n+1}}\boxminus\overline{K_{n+1}})^{n}_{A}$, $({K_{n+2}}\boxminus{K_{n+2}})^{n}_{A}$, $({K_{n+2}}\boxtimes\overline{K_{n+2}})^{n}_{A}$, and $({K_{n+1}}\boxtimes{K_{n+1}})^{n}_{A}$ have vertex-minors isomorphic to $nT_{2,n}$.

Proof.

(1) Let $V({K_{n+1}})=\{v_{i}:1\leq i\leq n+1\}$ and $V(\overline{K_{n+1}})=\{w_{i}:1\leq i\leq n+1\}$. The graph $({K_{n+1}}\boxminus\overline{K_{n+1}}-w_{1})*v_{1}$ is isomorphic to $T_{2,n}$.

(2) Let $\{v_{i}:1\leq i\leq n+2\}$ and $\{w_{i}:1\leq i\leq n+2\}$ be the vertex sets of two copies of ${K_{n+2}}$. The graph $(({K_{n+2}}\boxminus{K_{n+2}}-\{v_{1},w_{2}\})*v_{2}*w_{1})-\{w_{1}\}$ is isomorphic to $T_{2,n}$.

(3) Let $V({K_{n+2}})=\{v_{i}:1\leq i\leq n+2\}$ and $V(\overline{K_{n+2}})=\{w_{i}:1\leq i\leq n+2\}$. The graph $(({K_{n+2}}\boxtimes\overline{K_{n+2}}-\{w_{1},v_{2}\})*v_{1}*w_{2})-\{w_{2}\}$ is isomorphic to $T_{2,n}$.

(4) Let $\{v_{i}:1\leq i\leq n+1\}$ and $\{w_{i}:1\leq i\leq n+1\}$ be the vertex sets of two copies of ${K_{n+1}}$. The graph $({K_{n+1}}\boxtimes{K_{n+1}}-w_{1})*v_{1}$ is isomorphic to the graph obtained from $\overline{K_{n}}\boxminus\overline{K_{n}}$ by adding a vertex $v_{1}$ adjacent to all other vertices. Thus, the graph $({K_{n+1}}\boxtimes{K_{n+1}}-w_{1})*v_{1}*v_{2}*\cdots*v_{n+1}$ is isomorphic to $T_{2,n}$. ∎

In the following lemma, we show that if $A$ is not symmetric, then we obtain $P_{n}$ as a vertex-minor.

Lemma 4.11**.**

Let $n\geq 2$ be an integer. If $A=(\begin{smallmatrix}0&b\\ c&d\end{smallmatrix})$ is a [math]-$1$ matrix such that $b\neq c$, then both $({K_{1}}\boxminus{K_{1}})^{n}_{A}$ and $({K_{1}}\boxtimes{K_{1}})^{n+1}_{A}$ have vertex-minors isomorphic to $P_{n}$.

Proof.

If $d=0$, then $({K_{1}}\boxminus{K_{1}})^{n}_{A}$ is isomorphic to $K_{n}\boxslash\overline{K_{n}}$ and $(K_{1}\boxtimes K_{1})^{n+1}_{A}$ contains an induced subgraph isomorphic to $K_{n}\boxslash\overline{K_{n}}$. If $d=1$, then $({K_{1}}\boxminus{K_{1}})^{n}_{A}$ is isomorphic to $K_{n}\boxslash{K_{n}}$ and $({K_{1}}\boxtimes{K_{1}})^{n+1}_{A}$ contains an induced subgraph isomorphic to $K_{n}\boxslash{K_{n}}$. By Lemma 2.2, there is a vertex-minor isomorphic to $P_{2n-2}$ in both cases. ∎

Lemma 4.12**.**

Let $n\geq 2$ be an integer. If $A=(\begin{smallmatrix}0&0\\ 0&1\end{smallmatrix})$, then all of $(K_{n+2}\boxminus K_{n+2})^{n+1}_{A}$, $(K_{n+2}\boxminus\overline{K_{n+2}})^{n+1}_{A}$, $(K_{n+2}\boxtimes K_{n+2})^{n+1}_{A}$, and $(K_{n+2}\boxtimes\overline{K_{n+2}})^{n+1}_{A}$ have vertex-minors isomorphic to $nT_{2,n}$.

Proof.

Let $H$ be the one of $K_{n+2}\boxminus K_{n+2}$, $K_{n+2}\boxminus\overline{K_{n+2}}$, $K_{n+2}\boxtimes K_{n+2}$, or $K_{n+2}\boxtimes\overline{K_{n+2}}$ and let $G=H^{n+1}_{A}$. Let $X_{1}$, $Y_{1}$ be the sets of vertices of the first copy of $H$ in $G$ where $X_{1}$ denotes the set of vertices in $K_{n+2}$ and $Y_{1}$ denotes the other vertices. Let $w$ be a vertex in $Y_{1}$.

Then $G*w$ contains an induced subgraph isomorphic to $(K_{n+2}\boxminus K_{n+2})^{n}_{B}$, $(K_{n+2}\boxminus\overline{K_{n+2}})^{n}_{B}$, $(K_{n+2}\boxtimes K_{n+2})^{n}_{B}$, or $(K_{n+2}\boxtimes\overline{K_{n+2}})^{n}_{B}$ for $B=(\begin{smallmatrix}0&0\\ 0&0\end{smallmatrix})$. By Lemma 4.10, it has a vertex-minor isomorphic to $nT_{2,n}$. ∎

Lemma 4.13**.**

Let $n\geq 2$ be an integer. If $A=(\begin{smallmatrix}0&1\\ 1&d\end{smallmatrix})$ for some $d\in\{0,1\}$, then all of $(K_{n+2}\boxminus K_{n+2})^{n+2}_{A}$, $(K_{n+2}\boxminus\overline{K_{n+2}})^{n+2}_{A}$, $(K_{n+2}\boxtimes K_{n+2})^{n+2}_{A}$, and $(K_{n+2}\boxtimes\overline{K_{n+2}})^{n+2}_{A}$ have vertex-minors isomorphic to $nT_{2,n}$.

Proof.

Let $H$ be the one of $K_{n+2}\boxminus K_{n+2}$, $K_{n+2}\boxminus\overline{K_{n+2}}$, $K_{n+2}\boxtimes K_{n+2}$, or $K_{n+2}\boxtimes\overline{K_{n+2}}$ and let $G=H^{n+2}_{A}$. There exists an induced subgraph $G^{\prime}$ of $G$ and an edge $xy$ of $G^{\prime}$ such that $G^{\prime}-x-y$ is isomorphic to $H^{n+1}_{A}$, $x$ is complete to the bottom part of copies of $H$, and anti-complete to the top part of copies of $H$, and $y$ is complete to the top part of copies of $H$, and is either complete or anti-complete to the bottom part of copies of $H$, because we can choose a vertex $x$ from the top part of $H$ and a neighbor $y$ is chosen from the bottom part in the same copy of $H$. Now, it is easy to see that $G^{\prime}*x*y*x-x-y$ is isomorphic to one of $(K_{n+2}\boxminus K_{n+2})^{n+1}_{B}$, $(K_{n+2}\boxminus\overline{K_{n+2}})^{n+1}_{B}$, $(K_{n+2}\boxtimes K_{n+2})^{n+1}_{B}$, and $(K_{n+2}\boxtimes\overline{K_{n+2}})^{n+1}_{B}$ for a matrix $B=(\begin{smallmatrix}0&0\\ 0&d\end{smallmatrix})$. By Lemmas 4.10 and 4.12, $G$ has a vertex-minor isomorphic to $nT_{2,n}$. ∎

5 Main proof

We are ready to prove our main theorem, restated below.

Theorem 1.3.

A vertex-minor ideal $\mathcal{C}$ has bounded depth-$2$ rank-brittleness if and only if

[TABLE]

and

[TABLE]

Before the proof, let us discuss why the two conditions in Theorem 1.3, $\{P_{1},P_{2},P_{3},\ldots\}\not\subseteq\mathcal{C}$ and $\{T_{2,1},2T_{2,2},3T_{2,3},\ldots\}\not\subseteq\mathcal{C}$, are incomparable. First we sketch the proof showing that no path contains $T_{2,3}$ as a vertex-minor. The tree $T_{2,3}$ is a tree having a vertex $v$ such that $T_{2,3}-v$ contains three components having linear rank-width $1$. (The definition of linear rank-width will be discussed in Section 6.) It implies that it has linear rank-width at least $2$, by a characterization of linear rank-width on trees, see [1, 2]. However, paths have linear rank-width $1$ and therefore no path contains $T_{2,3}$ as a vertex-minor. Thus, if $\mathcal{C}$ is the set of all vertex-minors of $P_{n}$ for all $n$, then $\mathcal{C}$ does not satisfy the first condition but satisfies the second condition. Secondly we claim that no $nT_{2,n}$ contains a long path as a vertex-minor. It is not difficult to see that $nT_{2,n}$ has depth-$3$ rank-brittleness at most $3$. However, $\{P_{1},P_{2},P_{3},\ldots\}$ has unbounded rank-depth [6], and thus unbounded depth-$3$ rank-brittleness. So, if $\mathcal{C}$ is the set of all vertex-minors of $nT_{2,n}$ for all $n$, then $\mathcal{C}$ does not satisfy the second condition but satisfies the first condition.

Now let us start the proof of Theorem 1.3. Our first lemma is to prove the forward implication. It is already known that $\{P_{1},P_{2},P_{3},\ldots\}$ has unbounded rank-depth [6] and therefore it has unbounded depth-$2$ rank-brittleness. Thus, to prove the forward implication, it is enough to show that $\{T_{2,1},2T_{2,2},3T_{2,3},4T_{2,4},\ldots\}$ has unbounded depth-$2$ rank-brittleness.

Lemma 5.1**.**

The class $\{T_{2,1},2T_{2,2},3T_{2,3},4T_{2,4},\ldots\}$ has unbounded depth-$2$ rank-brittleness.

Proof.

We claim that $nT_{2,n}$ has depth-$2$ rank-brittleness at least $n/2$. Suppose that $nT_{2,n}$ admits a $(m,2)$-decomposition $(T,\sigma)$ with $m<n/2$. Then $T$ has a root $r$ from which every leaf is within distance at most $2$, and we may assume that $r$ is not a leaf. By subdividing an edge if necessary, we may further assume that no leaf is adjacent to $r$.

Let $r_{1}$, $r_{2}$, $\ldots$, $r_{m}$ be the neighbors of $r$. We color each vertex $v$ of $nT_{2,n}$ by $i\in\{1,2,\ldots,m\}$ if the component of $T-r$ containing $\sigma(v)$ has $r_{i}$. An edge of $nT_{2,n}$ is colorful if its ends have distinct colors. Let $C_{1}$, $C_{2}$, $\ldots$, $C_{n}$ be the components of $nT_{2,n}$.

Suppose that a component $C_{i}$ is fully contained in $X_{j}$ for some $j$. Then, since $C_{i}$ contains an induced matching of size $n$, the width of $r_{j}$ has to be at least $n$. This contradicts the assumption that $(T,\sigma)$ has width less than $n/2$. Thus, we may assume that no component $C_{i}$ is fully contained in some $X_{j}$. So every component $C_{i}$ has a colorful edge and therefore $nT_{2,n}$ has a set $F$ of $n$ colorful edges in distinct components.

Let $X$ be a subset of $\{1,2,\ldots,m\}$ chosen uniformly at random. A colorful edge of $nT_{2,n}$ is $X$-colorful if one end has a color in $X$ and the other end has a color not in $X$. Then by the linearity of expectation, the expected number of $X$-colorful edges in $F$ is $n/2$. This means that there exists $X$ such that there are at least $n/2$ $X$-colorful edges in distinct components of $nT_{2,n}$ and so the width of $r$ is at least $n/2$, contradicting the assumption on $(T,\sigma)$. ∎

The following proposition proves the backward implication of Theorem 1.3.

Proposition 5.2**.**

For every integer $n\geq 2$, there exists an integer $N:=d(n)$ such that every graph $G$ of depth-$2$ rank-brittleness at least $N$ contains a vertex-minor isomorphic to $P_{n}$ or $nT_{2,n}$.

Proof.

Let $\sigma$ be the function defined in Proposition 4.5 and let $f$, $g$ be the functions defined in Lemma 4.9. Let $m:=\max(n,f(n+2,n+2),g(n+2,n+2))$. and let $d(n):=\sigma(m)$. By Proposition 4.5, $G$ has a vertex-minor $G^{\prime}$ satisfying one of the following:

1. (i)

$V(G^{\prime})=X_{1}^{*}\cup X_{2}^{*}\cup\cdots\cup X_{m}^{*}\cup Q^{*}$ for disjoint sets $X_{1}^{*}$, $X_{2}^{*},\ldots,X_{m}^{*},Q^{*}$ of $m$ vertices such that each $X_{i}^{*}$ is a clique in $G^{\prime}$, $G^{\prime}[X_{1}^{*}\cup X_{2}^{*}\cup\cdots\cup X_{m}^{*}]$ is isomorphic to $mK_{m}$, and either $K_{m}\boxminus\overline{K_{m}}$ or $K_{m}\boxtimes\overline{K_{m}}$ is isomorphic to all $G^{\prime}[X_{i}^{*}\cup Q^{*}]$. 2. (ii)

$V(G^{\prime})=X_{1}^{*}\cup X_{2}^{*}\cup\cdots\cup X_{m}^{*}\cup Y_{1}^{*}\cup Y_{2}^{*}\cup\cdots\cup Y_{m}^{*}$ for disjoint sets $X_{1}^{*}$, $X_{2}^{*},\ldots,X_{m}^{*},Y_{1}^{*},Y_{2}^{*},\ldots,Y_{m}^{*}$ of $m$ vertices such that each $X_{i}^{*}$ is a clique in $G^{\prime}$, $G^{\prime}[X_{1}^{*}\cup X_{2}^{*}\cup\cdots\cup X_{m}^{*}]$ is isomorphic to $mK_{m}$, and one of $K_{m}\boxminus\overline{K_{m}}$, $K_{m}\boxminus K_{m}$, $K_{m}\boxtimes\overline{K_{m}}$, and $K_{m}\boxtimes K_{m}$ is isomorphic to all $G^{\prime}[X_{i}^{*}\cup Y_{i}^{*}]$. 3. (iii)

$G^{\prime}$ is isomorphic to $P_{m}$.

If (i) holds, then by Lemmas 4.6 and 4.7, $G^{\prime}$ has a vertex-minor isomorphic to $P_{3m-1}$ or $P_{4m-5}$. So if (i) or (iii) holds, then $G$ has a vertex-minor isomorphic to $P_{n}$. If (ii) holds, then by Lemma 4.9, $G^{\prime}$ has an induced subgraph isomorphic to one of

[TABLE]

for some [math]-$1$ matrix $A=(\begin{smallmatrix}0&b\\ c&d\end{smallmatrix})$. By Lemmas 4.10, 4.11, 4.12, and 4.13, $G^{\prime}$ has a vertex-minor isomorphic to $P_{n}$ to $nT_{2,n}$. ∎

6 Rank-depth and linear rank-width

By Theorem 1.3, for a fixed positive integer $n$, $nP_{5}$-vertex-minor free graphs have bounded depth-$2$ rank-brittleness, and thus have bounded rank-depth. We will show that they have bounded linear rank-width. Indeed, we will show that graphs of bounded rank-depth have bounded linear rank-width. This was also proved by Ganian, Hliněný, Nešetřil, Obdržálek, and Ossona de Mendez [11, Proposition 3.4] in terms of shrub-depth and linear clique-width, but our proof provides an explicit bound.

First let us review the definition of linear rank-width [10, 15, 23]. For a graph $G$, an ordering $(x_{1},\ldots,x_{n})$ of the vertex set $V(G)$ is called a linear layout of $G$. If $\lvert V(G)\rvert\geq 2$, then the width of a linear layout $(x_{1},\ldots,x_{n})$ of $G$ is defined as $\displaystyle\max_{1\leq i\leq n-1}\rho_{G}(\{x_{1},\ldots,x_{i}\})$, and if $\lvert V(G)\rvert=1$, then the width is defined to be [math]. The linear rank-width of $G$, denoted by $\operatorname{lrw}(G)$, is defined as the minimum width over all linear layouts of $G$. It is easy to see that if $H$ is a vertex-minor of $G$, then $\operatorname{lrw}(H)\leq\operatorname{lrw}(G)$.

Proposition 6.1**.**

For a graph $G$, $\operatorname{lrw}(G)\leq\operatorname{rd}(G)^{2}$.

Proof.

If $G$ has $1$ vertex, then $\operatorname{lrw}(G)=\operatorname{rd}(G)=0$. So, we may assume that $G$ has at least $2$ vertices. Let $k=\operatorname{rd}(G)$, and let $(T,\sigma)$ be a $(k,k)$-decomposition of $G$. Let $r$ be a node of $T$ within distance at most $k$ from every node of $T$.

Let $v_{1},v_{2},\ldots,v_{m}$ be a DFS ordering of $T$. Let $n=\lvert V(G)\rvert$. Let $w_{1},w_{2},\ldots,w_{n}$ be an ordering of the vertices of $G$ such that for all $1\leq i<j\leq n$, $\sigma(i)$ appears before $\sigma(j)$ in the DFS ordering $v_{1},v_{2},\ldots,v_{m}$ of $T$. We claim that $(w_{1},w_{2},\ldots,w_{n})$ has width at most $k^{2}$. Let $i\in\{1,2,\ldots,m\}$, $A_{i}:=\{v_{1},\ldots,v_{i}\}$, $B_{i}:=V(T)\setminus A_{i}$, $A_{i}^{\prime}=\{v\in v(G):\sigma(v)\in A_{i}\}$, and $B_{i}^{\prime}=\{v\in V(G):\sigma(v)\in B_{i}\}$. By the property of the depth-first search, $T$ has a path $P_{i}$ from $r$ consisting of nodes in $A_{i}$ such that for each node $w$ in $B_{i}$, the first vertex in $A_{i}$ in the path from $w$ to $r$ is on $P_{i}$.

As $T$ has radius at most $k$, we can take $P_{i}$ to have length at most $k-1$.

For $w\in V(P_{i})$, let $X_{w}$ be the set of all vertices $x$ of $G$ mapped to a node $\sigma(x)$ in $B_{i}$ such that $w$ is the first vertex in $A_{i}$ in the path from $\sigma(x)$ to $r$ is on $P_{i}$.

Since $(T,\sigma)$ has width at most $k$, the cut-rank of $X_{w}$ is at most $k$. As $B_{i}^{\prime}=\bigcup_{w\in V(P_{i})}X_{w}$, we deduce that $\rho_{G}(B_{i}^{\prime})\leq\sum_{w\in V(P_{i})}\rho_{G}(X_{w})\leq k^{2}$ by the submodularity of the cut-rank function. This implies that the width of the linear layout is at most $k^{2}$. ∎

Corollary 1.4.

For every positive integer $n$, graphs with no vertex-minors isomorphic to $nP_{5}$ have bounded depth-$2$ rank-brittleness, bounded rank-depth, and bounded linear rank-width.

Proof.

Let $\mathcal{C}$ be the class of $nP_{5}$-vertex-minor free graphs. Then $P_{6n}\notin\mathcal{C}$ and $nT_{2,n}\notin\mathcal{C}$. Thus, by Theorem 1.3, $\mathcal{C}$ has bounded depth-$2$ rank-brittleness, and thus bounded rank-depth. By Proposition 6.1, it also has bounded linear rank-width. ∎