On $k$-Connected $\Gamma$-Extensions of Binary Matroids

TL;DR

This paper characterizes when the $ ext{Gamma}$-extension operation preserves $k$-connectedness in binary matroids and provides conditions to connect disconnected matroids through this operation.

Contribution

It offers necessary and sufficient conditions for preserving $k$-connectedness and connectivity in binary matroids under the $ ext{Gamma}$-extension operation.

Findings

Conditions for $k$-connectedness preservation

Criteria for connecting disconnected matroids

Generalization of point-addition in graphs

Abstract

Slater introduced the point-addition operation on graphs to classify 4-connected graphs. The $\Gamma$-extension operation on binary matroids is a generalization of the point-addition operation. In this paper, we obtain necessary and sufficient conditions to preserve $k$-connectedness of a binary matroid under the $\Gamma$-extension operation. We also obtain a necessary and sufficient condition to get a connected matroid from a disconnected binary matroid using the $\Gamma$-extension operation.

Full text

This research is supported by DST-SERB, Government of India under the project SR/S4/MS:750/12

Keywords: binary matroid, splitting, $k$-connected, $\Gamma$-extension Mathematics Subject Classification: 05B35

1. Introduction

We refer to [9] for standard terminology in graphs and matroids. The matroids considered here are loopless and coloopless. Slater [12] introduced the point-addition operation on graphs and used it to classify $4$-connected graphs. Azanchiler [1] extended this operation to binary matroids as follows:

Definition 1.1**.**

[1]** Let $M$ be a binary matroid with ground set $S$ and standard matrix representation $A$ over $GF(2).$ Let $X=\{x_{1},x_{2},\dots,x_{m}\}\subset S$ be an independent set in $M$ and let $\Gamma=\{\gamma_{1},\gamma_{2},\dots,\gamma_{m}\}$ be a set such that $S\cap\Gamma=\phi.$ Suppose $A^{\prime}$ is the matrix obtained from the matrix $A$ by adjoining $m$ columns labeled by $\gamma_{1},\gamma_{2},...,\gamma_{m}$ such that the column labeled by $\gamma_{i}$ is same as the column labeled by $x_{i}$ for $i=1,2,\dots,m.$ Let $A^{X}$ be the matrix obtained by adjoining one extra row to $A^{\prime}$ which has entry 1 in the column labeled by $\gamma_{i}$ for $i=1,2\dots,m$ and zero elsewhere. The vector matroid of the matrix $A^{X},$ denoted by $M^{X},$ is called as the $\Gamma$-extension of $M$ and the transition from $M$ to $M^{X}$ is called as $\Gamma$-extension operation on $M.$

An example given at the end of the paper illustrates the definition. Note that the ground set of the matroid $M^{X}$ is $S\cup\Gamma$ and $M^{X}\setminus\Gamma=M.$ Therefore $M^{X}$ is an extension of $M.$ The $\Gamma$-extension operation is related to the splitting operation on binary matroids, which is defined by Shikare et al. [11], as follows:

Definition 1.2**.**

[11]** Let $M$ be a binary matroid with standard matrix representation $A$ over $GF(2)$ and let $Y$ be a non-empty set of elements of $M.$ Let $A_{Y}$ be the matrix obtained by adjoining one extra row to the matrix $A$ whose entries are 1 in the columns labeled by the elements of the set $Y$ and zero otherwise. The vector matroid of the matrix $A_{Y},$ denoted by $M_{Y},$ is called as the splitting matroid of $M$ with respect to $Y,$ and the transition from $M$ to $M_{Y}$ is called as the splitting operation with respect to $Y.$

Let $M$ be a binary matroid with ground set $S$ and let $X=\{x_{1},x_{2},\dots,x_{m}\}$ be an independent set in $M.$ Obtain the extension $M^{\prime}$ of $M$ with ground set $S\cup\Gamma,$ where $\Gamma=\{\gamma_{1},\gamma_{2},\dots,\gamma_{m}\}$ is disjoint from $S,$ such that $\{x_{i},\gamma_{i}\}$ is a 2-circuit in $M^{\prime}$ for each $i.$ The matroid $M^{\prime}_{\Gamma}$ obtained from $M^{\prime}$ by splitting the set $\Gamma$ is the $\Gamma$-extension matroid $M^{X}.$

The splitting operation with respect to a pair of elements, which is a special case of Definition 1.2, was earlier defined by Raghunathan et al. [10] for binary matroids as an extension of the corresponding graph operation due to Fleischner [7].

Whenever we write $M^{X},$ it is assumed that $X$ is a non-empty independent set of the matroid $M.$

Azanchiler [1] characterized the circuits and the bases of the $\Gamma$-extension matroid $M^{X}$ in terms the circuits and bases of $M,$ respectively. Some results on preserving graphicness of $M$ under the $\Gamma$-extension operation are obtained in [2]. Borse and Mundhe [6] characterized the binary matroids $M$ for which $M^{X}$ is graphic for any independent set $X$ of $M.$

A $k$-separation of a matroid $M$ is a partition of its ground set $S$ into two disjoint sets $A$ and $B$ such that $min~{}\{\left|A\right|,\left|B\right|\}\geq k$ and $r(A)+r(B)-r(M)\leq k-1.$ A matroid $M$ is $k$-connected if it does not have a $(k-1)$-separation. Also, $M$ is connected if it is 2-connected.

In general, the splitting operation does not preserve the connectivity of a given matroid. Borse and Dhotre [4] provided a sufficient condition to preserve connectedness of a matroid while Borse [3] gave a sufficient condition to get a $k$-connected matroid from given the $(k+1)$-connected binary matroid, under the splitting with respect to a pair of elements. Borse and Mundhe [5], and Malwadkar et al. [8] gave two characterizations for getting a $k$-connected matroid from the given $(k+1)$-connected binary matroid by splitting with respect to any set of $k$ elements.

The $\Gamma$-extension operation also does not give $k$-connected matroid from the given $k$-connected binary matroid in general. Azanchiler [1] obtained sufficient conditions to preserve $2$-connectedness and 3-connectedness of a binary matroid under this operation.

In this paper, we obtain necessary and sufficient conditions to preserve $k$-connectedness under the $\Gamma$-extension operation for any integer $k\geq 2.$ We also give necessary and sufficient conditions to get a connected matroid from a disconnected binary matroid in terms of the $\Gamma$-extension operation.

2. Proofs

We need some lemmas.

Lemma 2.1**.**

[1*]** Let $M$ be a binary matroid with ground set $S$ and let $X$ be an independent set in $M.$ Suppose $M^{X}$ is the $\Gamma$-extension of $M$ with ground set $S\cup\Gamma.$ Let $r$ and $r^{\prime}$ be the rank functions of $M$ and $M^{X},$ respectively. Then

(i) $\Gamma$ is independent in $M^{X};$

(ii) $r^{\prime}(A)=r(A)$ if $A\subset S;$

(iii) $r^{\prime}(A)\geq r(S\cap A)+1$ if $A$ intersects $\Gamma;$

(iv) $r^{\prime}(M^{X})=r(M)+1.$

Lemma 2.2**.**

[1*]** Let $M$ be a binary matroid with ground set $S$ and let $X$ be an independent set in $M.$ Then $Z\subset S\cup\Gamma$ is a circuit of $M^{X}$ if and only if one of the following conditions holds:

(i) $Z$ is a circuit of $M;$

(ii) $Z=\{x_{i},x_{j},\gamma_{i},\gamma_{j}\}$ for some distinct elements $x_{i},x_{j}$ of $X$ and the corresponding elements $\gamma_{i},\gamma_{j}$ of $\Gamma;$

(iii) $Z=J\cup(D-X_{J}),$ where $J\subset\Gamma$ with $|J|$ even and $D$ is a circuit of $M$ containing the set $X_{J}=\{x_{i}\in X\colon\gamma_{i}\in J\}.$*

Lemma 2.3** ([9], pp 273).**

Let $M$ be a $k$-connected matroid with at least $2(k-1)$ elements. Then every circuit and every cocircuit of $M$ contains at least $k$ elements.

The next lemma is a consequence of [9, Proposition 2.1.6].

Lemma 2.4**.**

[3]** Let $M$ be a matroid with ground set $S$ and let $Y\subset S$ such that $r(M\setminus Y)=r(M)-1.$ Then $Y$ contains a cocircuit of $M.$

The following result follows immediately from Lemma 2.3 and Lemma 2.4.

Corollary 2.5**.**

Let $M$ be a $k$-connected matroid with ground set $S$ such that $|S|\geq 2(k-1).$ Then $r(M\backslash Y)=r(M)$ for any $Y\subset S$ with $|Y|<k.$

We now give necessary and sufficient conditions to obtain a $k$-connected matroid from the given $k$-connected binary matroid as follows.

Theorem 2.6**.**

Let $k\geq 2$ be an integer and $M$ be a $k$-connected binary matroid with at least $2(k-1)$ elements and $X$ be an independent set in $M.$ Then the $\Gamma$-extension matroid $M^{X}$ is $k$-connected if and only if $|X|\geq k$ and $2\leq k\leq 4$.

Proof.

Suppose $|X|\geq k$ and $2\leq k\leq 4.$ We prove that $M^{X}$ is $k$-connected. The ground set of $M^{X}$ is $S\cup\Gamma,$ where $\Gamma$ is disjoint from the ground set $S$ of $M.$ Since $|\Gamma|=|X|,$ $|\Gamma|\geq k.$ By Lemma 2.1(i), $\Gamma$ is independent in $M^{X}.$ Suppose $r$ and $r^{\prime}$ denote the rank functions of $M$ and $M^{X},$ respectively. Assume that $M^{X}$ is not $k$-connected. Then $M^{X}$ has a $(k-1)$-separation $(A,B).$ Therefore $A$ and $B$ are non-empty disjoint subsets of $S\cup\Gamma$ such that $S\cup\Gamma=A\cup B$ and further,

min $\{\left|A\right|,\left|B\right|\}\geq k-1~{}~{}~{}~{}{\rm and}$

$r^{\prime}(A)+r^{\prime}(B)-r^{\prime}(M^{X})\leq k-2.~{}~{}~{}~{}~{}\ldots\ldots(1)$

As $A$ and $B$ are non-empty, each of them intersects $S$ or $\Gamma$ or both. We consider the three cases depending on whether $A$ intersect only $S$ or only $\Gamma$ or both and obtain a contradiction in each of these cases. Case (i). $A$ intersects only $\Gamma.$ As $A\subset\Gamma,$ $B=S\cup(\Gamma-A).$ Since $\Gamma$ is independent, $A$ is independent in $M^{X}.$ Consequently, $r^{\prime}(A)=|A|\geq k-1.$ Suppose $A\neq\Gamma.$ Then, by Lemma 2.1(iii) and (iv), $r^{\prime}(B)\geq r(S)+1=r(M)+1=r^{\prime}(M^{X}).$ Therefore $r^{\prime}(B)=r^{\prime}(M^{X}).$ Hence $r^{\prime}(A)+r^{\prime}(B)-r^{\prime}(M^{X})\geq k-1,$ which contradicts (1). Therefore $A=\Gamma.$ Hence $B=S$ and $r^{\prime}(A)=|\Gamma|\geq k.$ By Lemma 2.1(ii) and (iv), $r^{\prime}(B)=r^{\prime}(S)=r(S)=r(M)=r^{\prime}(M^{X})-1.$ Therefore $r^{\prime}(A)+r^{\prime}(B)-r^{\prime}(M^{X})\geq k-1,$ which is a contradiction to (1). Case (ii). $A$ intersects only $S.$ As $A\cap\Gamma=\phi,$ $A\subset S$ and $B=(S-A)\cup\Gamma.$ Therefore, by Lemma 2.1(i) and (ii), $r^{\prime}(A)=r(A)$ and $r^{\prime}(B)\geq r^{\prime}(\Gamma)=|\Gamma|\geq k.$ Suppose $|S-A|\leq k-2.$ Then, by Corollary 2.5, $r(A)=r(M)$. Consequently, by Lemma 2.1(iv),

$r^{\prime}(A)+r^{\prime}(B)-r^{\prime}(M^{X})=r(A)+r^{\prime}(B)-(r(M)+1)\geq r^{\prime}(B)-1\geq k-1,$

which is a contradiction to (1). Hence $|S-A|\geq k-1.$ By Lemma 2.1 (ii) and (iii), $r(S-A)=r^{\prime}(S-A)\leq r^{\prime}(B)-1.$ Therefore, by Inequality (1),

$r(A)+r(S-A)-r(M)\leq r^{\prime}(A)+r^{\prime}(B)-1-r(M^{X})+1\leq k-2.$

This shows that $A$ and $S-A$ gives a $(k-1)$-separation of $M,$ which is a contradiction to fact that $M$ is $k$-connected. Case (iii). $A$ intersects both $S$ and $\Gamma.$ Let $S_{1}=A\cap S$ and $\Gamma_{1}=A\cap\Gamma.$ Since $B\neq\phi,$ it intersects $S$ or $\Gamma.$ If $B$ intersects only $S$ or only $\Gamma,$ then we get a contradiction by interchanging roles of $A$ and $B$ in Case (i) and Case (ii). Therefore $B$ intersects both $S$ and $\Gamma.$ Let $S_{2}=B\cap S$ and $\Gamma_{2}=B\cap\Gamma.$ Then $S_{i}\neq\phi$ and $\Gamma_{i}\neq\phi$ for $i=1,2.$ By Lemma 2.1(ii) and (iii), $r(S_{1})=r^{\prime}(S_{1})\leq r^{\prime}(A)-1$ and $r(S_{2})=r^{\prime}(S_{2})\leq r^{\prime}(B)-1.$ By (1),

$r(S_{1})+r(S_{2})-r(M)\leq r^{\prime}(A)-1+r^{\prime}(B)-1-r^{\prime}(M^{X})+1\leq k-3.$

Hence, if $|S_{1}|\geq k-2$ and $|S_{2}|\geq k-2,$ then $(S_{1},S_{2})$ gives a $(k-2)$-separation of $M,$ a contradiction to fact that $M$ is $k$-connected. Consequently, $|S_{1}|\leq k-3$ or $|S_{2}|\leq k-3.$

Suppose $|S_{1}|\leq k-3.$ As $k\leq 4$ and $1\leq|S_{1}|,$ $k=4$ and $|S_{1}|=k-3=1.$ Thus $A$ contains exactly one element, say $x,$ of $M.$ Further, $|A|\geq k-1=4-1=3.$ We claim that $r^{\prime}(A)\geq 3.$ Suppose $r^{\prime}(A)\leq 2.$ Then $A$ contains a circuit $C$ of $M^{X}$ such that $|C|\leq 3.$ Since $\Gamma$ is independent in $M^{X},$ $C$ is not a subset of $\Gamma.$ Therefore $C$ contains $x$ and $C-\{x\}\subset A-\{x\}\subset\Gamma.$ In the last row of the matrix $A^{X}$ which represents the matroid $M^{X},$ the columns corresponding to the elements of $\Gamma$ have entries 1 and rest of the entries in that row are zero. As $C$ is a circuit, the sum of the columns of $A^{X}$ corresponding to the elements of $C$ is zero over GF(2). This implies that $C$ contains at least two elements of $\Gamma.$ Hence $C=\{x,\gamma_{1},\gamma_{2}\}$ for some $\gamma_{1},\gamma_{2}\in\Gamma.$ Let $x_{1}$ and $x_{2}$ be elements of the matroid $M$ corresponding to $\gamma_{1}$ and $\gamma_{2},$ respectively. By Lemma 2.2(ii), $C_{1}=\{x_{1},x_{2},\gamma_{1},\gamma_{2}\}$ is a circuit in $M^{X}.$ Since $M^{X}$ is a binary matroid, the symmetric difference $C\Delta C_{1}=\{x,x_{1},x_{2}\}$ of the circuits $C$ and $C_{1}$ contains a circuit, say $C_{2},$ of $M^{X}.$ Hence $C_{2}$ is a circuit in $M^{X}\setminus\Gamma=M$ such that $|C_{2}|\leq 3=4-1=k-1,$ a contradiction by Lemma 2.3. Hence $r^{\prime}(A)\geq 3.$ Since $|S_{1}|\leq k-3,$ by Corollary 2.5, $r(S_{2})=r(S-S_{1})=r(M).$ Therefore, by Lemma 2.1(iii), $r^{\prime}(B)\geq r^{\prime}(S_{2})+1=r(S_{2})+1=r(M)+1=r^{\prime}(M^{X}).$ Therefore $r^{\prime}(B)=r^{\prime}(M^{X}).$ Hence $r^{\prime}(A)+r^{\prime}(B)-r^{\prime}(M^{X})=r^{\prime}(A)\geq 3=k-1,$ a contradiction to (1).

Suppose $|S_{2}|\leq k-3.$ Then, as in the above paragraph, we see that $r^{\prime}(B)\geq 3=k-1$ and $r^{\prime}(A)=r^{\prime}(M^{X})$ and so $r^{\prime}(A)+r^{\prime}(B)-r^{\prime}(M^{X})=r^{\prime}(B)\geq k-1,$ a contradiction to (1).

Thus we get contradictions in Cases (i), (ii) and (iii). Therefore $M^{X}$ is $k$-connected.

Conversely, suppose $M^{X}$ is $k$-connected. The last row of the matrix $A^{X},$ which represents $M^{X},$ has 1’s in the columns corresponding to the set $\Gamma$ and zero elsewhere. Hence $\Gamma$ contains a cocircuit of $M^{X}.$ By Lemma 2.3, $|\Gamma|\geq k$ and so $|X|=|\Gamma|\geq k.$ By Lemma 2.2(ii), $M^{X}$ contains a 4-circuit. Therefore, by Lemma 2.3, $k\leq 4.$ This completes the proof. ∎

We now give a necessary and sufficient condition to get a connected matroid $M^{X}$ from the disconnected matorid $M.$ If $X$ is disjoint from a component $D$ of $M,$ then it follows from Lemma 2.2 that $D$ is a component of $M^{X}$ also. Therefore to get a connected matroid $M^{X}$ from the disconnected matroid $M,$ it is necessary that $X$ intersects every component of $M.$ In the following theorem, we prove that this obvious necessary condition is also suffcient.

Theorem 2.7**.**

Let $M$ be a disconnected binary matroid and let $X$ be an independent set in $M.$ Then $M^{X}$ is connected if and only if every component of $M$ intersects $X.$

Proof.

Let $M_{1},M_{2},\dots,M_{r}$ be the components of $M.$ Suppose each $M_{i}$ intersects $X$. Let $S$ be the ground set of $M.$ Then the ground set of $M^{X}$ is $S\cup\Gamma,$ where $S\cap\Gamma=\phi.$ Since each $M_{i}$ is connected in $M$ and $M^{X}\setminus\Gamma=M,$ each $M_{i}$ is connected in $M^{X}$ too. Therefore each $M_{i}$ is contained in a component of $M^{X}.$ We show that all $M_{i}$ are contained in a single component of $M^{X}.$ Since $M$ is disconnected, it has at least two components and so $r\geq 2.$ Let $D$ be a component of $M^{X}$ containing $M_{1}$ and let $j\in\{2,3,\dots,r\}.$ Suppose $X$ contains an element $x_{1}$ of $M_{1}$ and an element $x_{j}$ of $M_{j}.$ Suppose $\gamma_{1}$ and $\gamma_{j}$ are elements of $\Gamma$ corresponding to $x_{1}$ and $x_{j},$ respectively. Then, by Lemma 2.2(ii), $C=\{x_{1},x_{j},\gamma_{1},\gamma_{j}\}$ is a 4-circuit in $M^{X}.$ As $C$ contains an element of the component $D$ of $M^{X},$ $C$ is contained in $D.$ Therefore $D$ contains the element $x_{j}$ of $M_{j}.$ Consequently, $M_{j}$ is contained in $D.$ Thus all components of $M$ are contained in $D.$ Therefore $S\subset D.$ Let $\gamma$ be an arbitrary member of $\Gamma$ and let $x$ be the member of $X$ corresponding to $\gamma.$ Then, by Lemma 2.2(ii), $\gamma$ and $x$ belong to a 4-circuit, say $Z,$ of $M^{X}.$ As $x\in Z\cap D,$ $Z\subset D$ and so $\gamma\in D.$ Therefore $\Gamma\subset D.$ Consequently, $D$ is the only component of $M^{X}.$ Hence $M^{X}$ is connected.

The converse readily follows from the discussion prior to the statement of the theorem. ∎

Example 2.8. We illustrate Theorem 2.6 by using the Fano matroid $F_{7}.$ The ground set of $F_{7}$ is $\{1,2,3,4,5,6,7\}$ and the standard matrix representation of $F_{7}$ over $GF(2)$ is as follows:

$A=$1 234567( 1000111 ) 01010110011101 . Let $X=\{1,2\}$ and $Y=\{1,2,3\}.$ Then $X$ and $Y$ are independent in $F_{7}.$ Further,

$A^{X}=$1 234567γ1γ2( 100011110 ) 010101101001110100 000000011 and $A^{Y}=$1 234567γ1γ2γ3( 1000111100 ) 01010110100011101001 0000000111 . Let $F_{7}^{X}$ and $F_{7}^{Y}$ be the vector matroids of $A^{X}$ and $A^{Y},$ respectively. It is well known that $F_{7}$ is 3-connected. One can check that $F_{7}^{Y}$ is 3-connected while $F_{7}^{X}$ is 2-connected but not 3-connected.