Connectivity of single-element coextensions of a binary matroid

TL;DR

This paper establishes a precise criterion for when adding a single element to a highly connected binary matroid preserves its connectivity level.

Contribution

It provides a necessary and sufficient condition for the connectivity of single-element coextensions of an n-connected binary matroid.

Findings

Characterization of connectivity preservation in coextensions

Necessary and sufficient condition derived

Applicable to binary matroids with high connectivity

Abstract

Given an $n$-connected binary matroid, we obtain a necessary and sufficient condition for its single-element coextensions to be $n$-connected.

Equations

$$r(A)+r(B-\{a\})-r(M)\le r^{\prime }(A)+r^{\prime }(B)-1-(r^{\prime }(M_T^{\prime })-1)\le n-2.$$

$$n-1\ge |Q|=|C^{\ast }|-|T|+1\ge (2n-2)-(n-1)+1=n,$$

$$|Q|\ge |(C_1^{\ast }\cup C_2^{\ast })-T|+1\ge |C_1^{\ast }|+|C_2^{\ast }|-|T|+1\ge 2n-(n-1)+1=n+2>n-1\ge |Q|,$$

$$|Q|=|C^{\ast }\Delta T|+1=|C^{\ast }|+|T|-2|C^{\ast }\cap T|+1\ge |T|+1=n>n-1\ge |Q|,$$

$$r(A-\{a\})+r(B)-r(M)\le r^{\prime }(A)-1+r^{\prime }(B)-(r^{\prime }(M_T^{\prime })-1)\le n-2.$$

$$r(A-\{a\})+r(B)-r(M)\le r^{\prime }(A)-1+r^{\prime }(B)-1-(r^{\prime }(M_T^{\prime })-1)=n-3.$$

Full text

Keywords: coextension, element splitting, point-splitting, binary matroids, $n$-connected

Subject Classification (2010): 05B35, 05C50

1. Introduction

For undefined terminologies, we refer to Oxley [6]. The point-splitting operation is a fundamental operation in respect of connectivity of graphs. It is used to characterize 3-connected graphs in the classical Tutte’s Wheel Theorem [9] and also to characterize 4-connected graphs by Slater [8]. This operation is defined as follows.

Definition 1.1** ([8]).**

Let $G$ be a graph with a vertex $v$ of degree at least $2n-2$ and let $T=\{vv_{1},vv_{2},\dots,vv_{n-1}\}$ be a set of $n-1$ edges of $G$ incident to $v$. Let $G_{T}^{\prime}$ be the graph obtained from $G$ by replacing $v$ by two adjacent vertices $u$ and $w$ such that $u$ is adjacent to $v_{1},v_{2},\dots,v_{n-1},$ and $w$ is adjacent to the vertices which are adjacent to $v$ except $v_{1},v_{2},\dots,v_{n-1}$. We say $G_{T}^{\prime}$ arises from $G$ by $n$-point splitting (see the following figure).

v$$u$$w$$v_{1}$$v_{2}$$v_{1}$$v_{2}$$G$$G_{T}^{\prime}

Slater [8] obtained the following result to characterize $4$-connected graphs.

Theorem 1.2** ([8]).**

Let $G$ be an $n$-connected graph and let $T$ be a set of $n-1$ edges incident to a vertex of degree at least $2n-2$. Then the graph $G^{\prime}_{T}$ is $n$-connected.

In this paper, we extend the above theorem to binary matroids.

Azadi [1] extended the $n$-point splitting operation on graphs to binary matroids as follows.

Definition 1.3** ([1]).**

Let $M$ be a binary matroid with standard matrix representation $A$ over the field $GF(2)$ and let $T$ be a subset of the ground set $E(M)$ of $M$. Let $A^{\prime}_{T}$ be the matrix obtained from $A$ by adjoining one extra row to matrix $A$ whose entries are 1 in the columns labeled by the elements of $T$ and 0 otherwise and also having one extra column labeled by $a$ with 1 in the last row and 0 elsewhere. Denote the vector matroid of $A^{\prime}_{T}$ by $M^{\prime}_{T}.$ We say that $M_{T}^{\prime}$ is obtained from $M$ by element splitting with respect to the set $T$.

For example, the following matrices $A$ and $A_{T}^{\prime}$ represent the Fano matroid $F_{7}$ and its element splitting matroid with respect to the set $T=\{1,2,3\}\subset E(F_{7}).$

A=\bordermatrix{~{}&1&2&3&4&5&6&7\cr~{}&1&0&0&1&1&0&1\cr~{}&0&1&0&1&1&1&0\cr~{}&0&0&1&1&0&1&1}, A^{\prime}_{T}=\bordermatrix{~{}&1&2&3&4&5&6&7&a\cr~{}&1&0&0&1&1&0&1&0\cr~{}&0&1&0&1&1&1&0&0\cr~{}&0&0&1&1&0&1&1&0\cr~{}&1&1&1&0&0&0&0&1}.

Given a graph $H$, let $M(H)$ denote the circuit matroid of $H$. A matroid $N$ is a single-element coextension of a matroid $M$ if $N/e=M$ for some element $e$ of $N$.

Definition 1.3 is an extension of Definition 1.1 as $M(G)_{T}^{\prime}=M(G_{T}^{\prime})$ for a set $T$ of edges incident to a vertex of a graph $G.$ Note that if $M$ is a binary matroid, then the element splitting matroid $M_{T}^{\prime}$ is also binary and it is a coextension of $M$ by the element $a$ as $M^{\prime}_{T}/a=M.$ In fact, we prove in Lemma 2.1 that every coextension of a binary matroid $M$ by a non-loop and non-coloop element is the element splitting matroid $M_{T}^{\prime}$ for some $T\subset E(M).$

Dalvi et al. [4, 5] characterized the graphic (cographic) matroids $M$ whose single-element coextensions $M_{T}^{\prime}$ are again graphic (cographic). Let $M$ be an $n$-connected binary matroid. Borse and Mundhe [3] obtained sufficient conditions for the matroid $M_{T}^{\prime}\backslash a$ to be $n$-connected. In this paper, we obtain a necessary and sufficient condition for $M_{T}^{\prime}$ to be $n$-connected. The following is the main theorem of the paper.

Main Theorem 1.4**.**

Let $n\geq 2$ be an integer and $M$ be an $n$-connected binary matroid with $|E(M)|\geq 2n-2$. Suppose $T\subset E(M)$ with $|T|=n-1$. Then $M^{\prime}_{T}$ is $n$-connected if and only if $|Q|\geq 2|Q\cap T|$ for every cocircuit $Q$ of $M$ intersecting $T.$

We also prove that Theorem 1.2 follows from Main Theorem 1.4 under a mild restriction.

Azadi [1] obtained the following result for $M_{T}^{\prime}$ to be $n$-connected, in terms of the circuits of $M$ containing an odd number of elements of $T$.

Theorem 1.5** ([1]).**

Let $n\geq 2$ be an integer and $M$ be an $n$-connected binary matroid with $|E(M)|\geq 2n-2$. Suppose $T\subset E(M)$ with $|T|=n-1$. Then $M^{\prime}_{T}$ is $n$-connected if and only if for any set $A\subset E(M)$ with $|A|=n-2$, there exists a circuit $C$ of $M$ containing an odd number of elements of $T$ and is contained in $E(M)-A.$

We provide an alternate shorter proof of Theorem 1.5 in the third section.

In Section 2, we provide some properties of $M_{T}^{\prime}.$ Main Theorem 1.4 is proved in Section 3. In the last section, we discuss consequences of Main Theorem 1.4 to the graphs.

2. Preliminaries

We prove below that the single-element coextension of a binary matroid $M$ by a non-loop and non-coloop element is nothing but an element splitting matroid $M_{T}^{\prime}$ for some $T\subset E(M)$.

Lemma 2.1**.**

Let $M$ and $N$ be binary matroids. Then $N$ is a coextension of $M$ by a non-loop and non-coloop element if and only if $N=M_{T}^{\prime}$ for some $T\subset E(M).$

Proof.

Suppose $N=M_{T}^{\prime}$ for some $T\subset E(M)$. Then the ground set of $N$ is $E(M)\cup\{a\}$ and $N/a=M$. Hence $N$ is a coextension of $M$ by the element $a.$ Let $A$ be the standard matrix representation of $M$ over $GF(2)$. By Definition 1.3, in the matrix $A_{T}^{\prime}$ of $M_{T}^{\prime},$ the column labeled by $a$ has 1 in the last row and 0 elsewhere, and the columns labeled by the elements of $T$ have 1 in the last row. This shows that $a$ is neither a loop nor a coloop of $N.$

Conversely, suppose $N$ is a coextension of $M$ by a non-loop and non-coloop element $a.$ Let $T_{1}$ be a cocircuit of $N$ containing $a$ and let $T=T_{1}-\{a\}$. Then $T$ is a non-empty subset of $E(M)$. We can write the standard matrix representation $B$ of $N$ such that the column of $B$ labeled by $a$ has entry 1 in the last row and 0 elsewhere. Since $T_{1}$ is a cocircuit of $N$, the last row of $B$ contains 1 in the columns corresponding to $T_{1}$ and 0 elsewhere. Let $C$ be the matrix obtained from $B$ by deleting the last row and the column corresponding to $a$. Then $M[C]=N/a=M$. Thus $B$ can be obtained from $C$ by adding one extra row which has entries 1 below the elements corresponding to $T$ and then adding a column labeled by $a$ which has entry 1 in the last row and 0 elsewhere. Therefore, by Definition 1.3, $B=C_{T}^{\prime}$. Hence $N=M[B]=M[C_{T}^{\prime}]=M_{T}^{\prime}$. ∎

Henceforth, we use the notation $M_{T}^{\prime}$ for a single-element coextension of a binary matroid $M.$

We need the following results.

Lemma 2.2** ([1]).**

Let $M$ be a binary matroid and $T\subseteq E(M)$. If $\mathcal{C}$ is the collection of circuits of $M$, then every circuit of $M^{\prime}_{T}$ belongs to one of the following type.

1. (i).

$\mathcal{C}_{1}=\{C\in\mathcal{C}\colon|C\cap T|~{}is~{}even\}$** 2. (ii).

$\mathcal{C}_{2}=\{C\cup\{a\}\colon C\in\mathcal{C}~{}and~{}contains~{}an~{}odd~{}number~{}of~{}elements~{}of~{}T\}$** 3. (iii).

$\mathcal{C}_{3}=set~{}of~{}minimal~{}members~{}of~{}\{C_{1}\cup C_{2}\colon C_{1},C_{2}\in\mathcal{C},~{}C_{1}\cap C_{2}=\emptyset~{}~{}and~{}C_{1}~{}and~{}C_{2}~{}each~{}\\ contains~{}an~{}odd~{}number~{}of~{}elements~{}of~{}T~{}such~{}that~{}C_{1}\cup C_{2}~{}does~{}not~{}contain~{}any~{}member~{}of~{}\mathcal{C}_{1}.\}$**

Lemma 2.3** ([2]).**

Let $M$ be a binary matroid. Suppose $r$ and $r^{\prime}$ are the rank functions of $M$ and $M^{\prime}_{T}$, respectively. If $A\subset E(M)\cup\{a\}$, then rank of $A$ is given by

1. (i).

$r^{\prime}(A)=r(A-\{a\})+1$* if $a\in A$.* 2. (ii).

$r^{\prime}(A)=r(A)+1$* if $a\notin A$ and $A$ contains a circuit $C$ of $M$ with $|C\cap T|$ odd.* 3. (iii).

$r^{\prime}(A)=r(A)$* if $a\notin A$ and $A$ does not contain any circuit $C$ of $M$ with $|C\cap T|$ odd.*

Corollary 2.4**.**

Let $M$ be a binary matroid and $T\subseteq E(M)$. Then $r^{\prime}(M_{T}^{\prime})=r(M)+1$.

Lemma 2.5** ([7]).**

Let $M$ be a binary matroid and $\mathcal{C^{*}}$ be the collection of cocircuits of $M$. Suppose $T\subseteq E(M)$ does not contain a cocircuit of $M$. Then every cocircuit of $M^{\prime}_{T}$ belongs to one of the following type.

1. (i).

$\mathcal{Q}_{1}^{*}=\{(C^{*}-T)\cup\{a\}\colon C^{*}\in\mathcal{C^{*}}~{}and~{}T~{}is~{}a~{}proper~{}subset~{}of~{}C^{*}\}$, 2. (ii).

$\mathcal{Q}_{2}^{*}=\{C^{*}\colon C^{*}\in\mathcal{C^{*}}\}$, 3. (iii).

$\mathcal{Q}_{3}^{*}=\{(C^{*}\Delta T)\cup\{a\}\colon C^{*}\in\mathcal{C^{*}},1\leq|C^{*}\cap T|<|T|~{}and~{}C^{*}~{}does~{}not~{}contain~{}D^{*}-T~{}for~{}any\\ ~{}D^{*}\in\mathcal{C^{*}}~{}and~{}T\subset D^{*}\},$** 4. (iv).

$\mathcal{Q}_{4}^{*}=\{((C^{*}_{1}\cup C^{*}_{2}\cup\dots\cup C^{*}_{k})-T)\cup\{a\}\colon k\geq 2,~{}C^{*}_{i}\in\mathcal{C^{*}},C^{*}_{i}\cap T\neq\emptyset,C^{*}_{i}~{}are~{}mutually~{}disjoint~{}\\ and~{}(C^{*}_{1}\cup C^{*}_{2}\cup\dots\cup C^{*}_{k})-T~{}does~{}not~{}contain~{}D^{*}-T~{}for~{}any~{}D^{*}\in\mathcal{C^{*}}~{}and~{}T\subset D^{*}\}$. 5. (v).

$\mathcal{Q}_{5}^{*}=\{T\cup\{a\}\}$.

3. Proofs

In this section, we prove Main Theorem 1.4 and also provide an alternate shorter proof of Theorem 1.5.

We need the following result.

Lemma 3.1** ([6], pp 296).**

If $n\geq 2$ and $M$ is an $n$-connected matroid with $|E(M)|\geq 2(n-1),$ then all circuits and all cocircuits of $M$ have at least $n$ elements.

Suppose $M$ is an $n$-connected binary matroid with $|E(M)|\geq 2(n-1)$ and $T\subset E(M).$ By Definition 1.3, there is a cocircuit of $M_{T}^{\prime}$ contained in $T\cup\{a\}.$ Therefore, if $|T|<n-1,$ then $M_{T}^{\prime}$ contains a cocircuit of size less than $n$ by Lemma 2.5 and hence $M_{T}^{\prime}$ is not $n$-connected by Lemma 3.1. Hence we assume that $|T|\geq n-1.$

We obtain below an obvious necessary condition for $M_{T}^{\prime}$ to be $n$-connected.

Lemma 3.2**.**

Let $n\geq 2$ be an integer and $M$ be an $n$-connected binary matroid with $|E(M)|\geq 2n-2$. Suppose $T\subset E(M)$ with $|T|=n-1$. If $M_{T}^{\prime}$ is $n$-connected, then $|Q|\geq 2|Q\cap T|$ for every cocircuit $Q$ of $M$ intersecting $T$.

Proof.

Suppose $M_{T}^{\prime}$ is $n$-connected. Assume that there is a cocircuit $Q$ of $M$ intersecting $T$ such that $|Q|<2|Q\cap T|.$ By Lemma 2.5 (iii), $Q\Delta T\cup\{a\}$ contains a cocircuit, say $X$, of $M^{\prime}_{T}$. Then $|X|\leq|Q\Delta T\cup\{a\}|=|Q|+|T|-2|Q\cap T|+1<|T|+1=n,$ a contradiction by Lemma 3.1. ∎

We now prove that the obvious necessary condition for $M_{T}^{\prime}$ to be $n$-connected stated in the above lemma is sufficient also.

Proposition 3.3**.**

Let $n\geq 2$ be an integer and $M$ be an $n$-connected binary matroid with $|E(M)|\geq 2n-2$. Suppose $T\subset E(M)$ with $|T|=n-1$. If $|Q|\geq 2|Q\cap T|$ for every cocircuit $Q$ of $M$ intersecting $T$, then $M^{\prime}_{T}$ is $n$-connected.

Proof.

Assume that $|Q|\geq 2|Q\cap T|$ for every cocircuit $Q$ of $M$ intersecting $T$. We proceed by contradiction. Suppose $M^{\prime}_{T}$ is not $n$-connected. Then there exists an $(n-1)$-separation $(A,B)$ of $M^{\prime}_{T}.$ Therefore

min$\{|A|,|B|\}\geq n-1$ and $r^{\prime}(A)+r^{\prime}(B)-r^{\prime}(M^{\prime}_{T})\leq n-2.$ ……$(*)$

Suppose $|A|\geq n$ and $|B|\geq n.$ Without loss of generality, we may assume that $a\in B$. By Lemma 2.3 and by $(*)$,

[TABLE]

Therefore $(A,B-\{a\})$ forms an $(n-1)$-separation of $M$, a contradiction.

Therefore $|A|=n-1$ or $|B|=n-1.$ We may assume that $|A|=n-1.$ Then $A$ is independent in $M$ by Lemma 3.1. Hence, by Lemma 2.2, $A$ is independent in $M^{\prime}_{T}$ also.

Claim: $A$ is a coindependent in $M^{\prime}_{T}$. Assume that $A$ is not coindependent in $M^{\prime}_{T}$. Then $A$ contains some cocircuit $Q$ of $M^{\prime}_{T}$. Therefore $|Q|\leq|A|=n-1.$ By Lemma 3.1, $Q$ is not a cocircuit of $M.$ Further, by Lemma 2.5, $Q$ does not belong to $\mathcal{Q}_{2}^{*}$. Hence $Q$ belongs to one of the four classes $\mathcal{Q}_{1}^{*}$, $\mathcal{Q}_{3}^{*}$, $\mathcal{Q}_{4}^{*}$ and $\mathcal{Q}_{5}^{*}.$

(1). Suppose $Q\in\mathcal{Q}_{1}^{*}.$ Then $Q=(C^{*}-T)\cup\{a\},$ where $C^{*}$ is a cocircuit of $M$ containing $T.$ Then, by hypothesis, $|C^{*}|\geq 2|C^{*}\cap T|=2|T|=2n-2$. Therefore

[TABLE]

a contradiction.

(2). Suppose $Q\in\mathcal{Q}_{4}^{*}.$ Then $Q=((C^{*}_{1}\cup C^{*}_{2}\cup\dots\cup C^{*}_{k})-T)\cup\{a\},$ where $k\geq 2$ and $C^{*}_{i}$ are mutually disjoint cocircuits of $M$ and each of them contains at least one element of $T.$ Since $M$ is $n$-connected, $|C^{*}_{i}|\geq n$ for each $i$ by Lemma 3.1. Hence, we have

[TABLE]

again a contradiction.

(3). Suppose $Q\in\mathcal{Q}_{3}^{*}.$ Then $Q=(C^{*}\Delta T)\cup\{a\},$ where $C^{*}$ is a cocircuit of $M$ intersecting $T.$ Hence

[TABLE]

a contradiction.

(4). Suppose $Q\in\mathcal{Q}_{5}^{*}$. So $Q=T\cup\{a\}.$ This gives $|Q|=n,$ a contradiction.

Thus in all the four cases, we get a contradiction. This proves the claim.

Therefore $A$ is independent and coindependent in the matroid $M^{\prime}_{T}.$ Hence $r^{\prime}(A)=|A|$ and $r^{\prime}(B)=r^{\prime}(M^{\prime}_{T}).$ This gives $n-1=|A|=r^{\prime}(A)=r^{\prime}(A)+r^{\prime}(B)-r^{\prime}(M^{\prime}_{T})\leq n-2,$ a contradiction. Thus we get a contradiction in each case. Therefore $M^{\prime}_{T}$ is $n$-connected. ∎

Main Theorem 1.4 follows obviously from Lemma 3.2 and Proposition 3.3.

For $2\leq n\leq 4$, we get the following weaker sufficient conditions for $M_{T}^{\prime}$ to be $n$-connected.

Corollary 3.4**.**

Let $n\in\{2,3,4\}$ and let $M$ be $n$-connected binary matroid. Suppose $T\subset E(M)$ with $|T|=n-1$. If $|Q|\geq 2n-2$ for every cocircuit $Q$ containing $T$, then $M^{\prime}_{T}$ is $n$-connected.

Proof.

Let $Q$ be a cocircuit of $M$ intersecting $T$. By Proposition 3.3, it is sufficient to prove that $|Q|\geq 2|Q\cap T|$. If $T\subseteq Q$, then $|Q|\geq 2n-2=2|T|=2|Q\cap T|$. Suppose $T\nsubseteq Q$. Then $|Q\cap T|<|T|=n-1$ and hence $|Q\cap T|\leq n-2$. Since $2\leq n\leq 4$, we have $2|Q\cap T|\leq 2(n-2)=2n-4\leq n.$ By Lemma 3.1, $|Q|\geq n$ and so $|Q|\geq 2|Q\cap T|$. ∎

We combine Main Theorem 1.4 and Theorem 1.5 and provide a shorter proof of Theorem 1.5.

Theorem 3.5**.**

Let $n\geq 2$ be an integer and $M$ be an $n$-connected binary matroid with $|E(M)|\geq 2n-2$. Suppose $T\subset E(M)$ with $|T|=n-1$. Then the following statements are equivalent.

1. (i).

$M^{\prime}_{T}$* is $n$-connected.* 2. (ii).

$|Q|\geq 2|Q\cap T|$* for every cocircuit $Q$ of $M$ intersecting $T.$* 3. (iii).

For any subset $A\subset E(M)$ with $|A|=n-2$, there exists a circuit $C$ of $M$ containing an odd number of elements of $T$ and is contained in $E(M)-A.$

Proof.

(i) $\implies$ (ii) follows from Lemma 3.2 and (ii) $\implies$ (i) follows from Proposition 3.3.

(i) $\implies$ (iii). Suppose (i) holds but (iii) does not hold. Then there is a subset $A$ of $E(M)$ with $|A|=n-2$ such that no circuit of $M$ containing an odd number of elements of $T$ is contained in $E(M)-A.$ Let $A^{\prime}=A\cup\{a\}$ and $B=E(M)-A.$ Then $|A^{\prime}|=n-1$ and $|B|\geq n-1.$ Let $r$ and $r^{\prime}$ be the rank function of $M$ and $M_{T}^{\prime}$, respectively. By Lemma 3.1, $A$ contains neither a cocircuit of $M$ nor a cocircuit of $M^{\prime}_{T}.$ Hence $r(B)=r(M)$ and $r^{\prime}(B)=r^{\prime}(M_{T}^{\prime})$. Also, by Lemma 2.3(iii), $r(B)=r^{\prime}(B).$ This gives $r(M)=r^{\prime}(M_{T}^{\prime})$, a contradiction by Corollary 2.4. Hence (i) implies (iii).

(iii) $\implies$ (i). Suppose (iii) holds but (i) does not hold. Then $M_{T}^{\prime}$ has an $(n-1)$-separation $(A,B).$ Therefore

min$\{|A|,|B|\}\geq n-1$ and $r^{\prime}(A)-r^{\prime}(B)-r^{\prime}(M^{\prime}_{T})\leq n-2$. ……$(*)$

Without loss of generality, assume that $a\in A.$ By Lemma 2.3(i), $r^{\prime}(A)=r(A-\{a\})+1$. If $|A|\geq n$, then, by $(*)$,

[TABLE]

Therefore $(A-\{a\},B)$ is an $(n-1)$-separation of $M$, a contradiction. Hence $|A|=n-1$. Then $|A-\{a\}|=n-2$. By (iii) and Lemma 2.3(ii), $r^{\prime}(B)=r(B)+1$. Therefore

[TABLE]

This shows that $(A,B)$ is an $(n-2)$-separation of $M,$ a contradiction. Thus (iii) implies (i). ∎

4. Consequences to Graphs

In this section, we prove that Proposition 3.3 is a matroid extension of Theorem 1.2.

We need the following result.

Theorem 4.1** ([6], pp. 328).**

For $n\geq 2$, let $G$ be a graph without isolated vertices and with at least $n+1$ vertices. Then the circuit matroid $M(G)$ is $n$-connected if and only if $G$ is $n$-connected and has no cycle with fewer than $n$ edges.

By Theorem 4.1, the circuit matroid $M(G)$ of an $n$-connected graph $G$ is not $n$-connected if $G$ contain a cycle of length less than $n$. Therefore we derive Theorem 1.2 from Proposition 3.3 by assuming that $G$ has girth at least $n$.

Theorem 4.2**.**

Suppose $G$ is an $n$-connected graph of girth at least $n$, where $n\geq 2$. Let $T$ be a set of $n-1$ edges incident to a vertex of degree at least $2n-2$ in $G$. Then the $n$-point splitting graph $G_{T}^{\prime}$ is $n$-connected.

Proof.

Let $M=M(G)$. Then $M_{T}^{\prime}=M(G_{T}^{\prime})$. We prove that $M_{T}^{\prime}$ is $n$-connected. By Theorem 4.1, $M$ is $n$-connected. Let $Q$ be a cocircuit of $M$ intersecting $T$. By Proposition 3.3, it is sufficient to prove that $|Q|\geq 2|Q\cap T|$. On the contrary, assume that $|Q|<2|Q\cap T|$. As $Q=(Q-T)\cup(Q\cap T)$, $|Q|=\cfrac{|Q|}{2}+\cfrac{|Q|}{2}=|Q-T|+|Q\cap T|$ and hence $|Q-T|<\cfrac{|Q|}{2}<|Q\cap T|.$ Let $u$ be the vertex of $G$ of degree at least $2n-2$ such that the edges of $G$ belonging to $T$ are incident to $u$. Since $Q$ is a cocircuit of $M(G)$, the graph $G-Q$ is disconnected and it has two components, say $C_{1}$ and $C_{2}$. We may assume that $C_{2}$ contains the vertex $u$. Let $Q\cap T=\{uu_{1},uu_{2},\dots,uu_{k}\}.$ Then $u_{1},u_{2},\dots,u_{k}$ are vertices of $C_{1}$. Let $v_{1},v_{2},\dots,v_{r}$ be the end vertices of the edges belonging to $Q-T$ in $C_{1}$. Then $r\leq|Q-T|<|Q\cap T|.$ Since $|Q|<2|Q\cap T|\leq 2|T|=2n-2$ and degree of $u$ is at least $2n-2$, there is at least one edge $uw$ incident to $u$ in $G-Q$. Then the edge $uw$ is in $C_{2}$. Let $A=\{v_{1},v_{2},\dots,v_{r},u\}$. Then $G-A$ is disconnected, leaving $u_{i}$ for some $i\in\{1,2,\dots,k\}$ in one component and the vertex $w$ is in an another component. However, $|A|=r+1\leq|Q\cap T|\leq|T|=n-1,$ a contradiction to the fact that $G$ is $n$-connected. Thus $M_{T}^{\prime}$ is $n$-connected. By Theorem 4.1, $G_{T}^{\prime}$ is $n$-connected. ∎

Corollary 4.3**.**

Let $G$ be a $3$-connected simple graph and $T$ be a set of two edges incident to a vertex of $G$ of degree at least four. Then the graph $G^{\prime}_{T}$ is $3$-connected.

We now prove that one can obtain a 3-regular, 3-connected graph from the given 3-connected simple graph by repeated applications of 3-point splitting operation.

Corollary 4.4**.**

A 3-regular, $3$-connected simple graph can be obtained from the given $3$-connected simple graph by a finite sequence of the 3-point splitting operation.

Proof.

Let $G$ be a $3$-connected simple graph. Then degree of every vertex of $G$ is at least three. Suppose $G$ contains a vertex $v$ of degree $k>3.$ Let $T=\{x,y\}$ be a set of two edges incident at $v$. By Corollary 4.3, $G^{\prime}_{T}$ is $3$-connected. The vertex of $v$ of $G$ is replaced by two vertices $v^{\prime}$ and $v^{\prime\prime}$ with degrees $3$ and $k-1,$ respectively in $G_{T}^{\prime}$. Thus one application of 3-point splitting on a vertex of degree $t>3$ results into a 3-connected graph with one additional vertex of degree less than $t$. By a finite sequence of 3-point splitting operation we can get a 3-connected graph with no vertex of degree greater than three. Clearly, this graph will be 3-regular. ∎