Lobachevskii Journal of Mathematics http://ljm.ksu.ru ISSN 1818-9962 Vol. 24, 2006, 3–12

© H. Azanchiler

Abstract. In this paper, we characterize the $n$-line splitting operation of graphs in terms of cycles of respective graphs and then extend this operation to binary matroids. In matroids, we call this operation an element-set splitting. The resulting matroid is called the es-splitting matroid. We characterize circuits of an es-splitting matroid. We also characterize the es-splitting matroid in terms of matrices. Also, we show that if $M$ is a connected binary matroid then the es-splitting matroid $M_X^e$ is also connected.

1. Introduction

In [9], Slater specified the $n$-point splitting and $n$-line splitting operations in graphs in the following way:

Let $G$ be a graph and $U$ be a vertex of $G$ with deg $U\ge 2n-2$. Let $G_1$ be the graph obtained from $G$ by replacing $U$ by two adjacent vertices $U_1$ and $U_2$, and if vertex $X$ is adjacent to $U$ in $G$, written $X$ adj $U$, then make $X$ adj $U_1$ or $X$ adj $U_2$ (but not both) such that deg $U_1\ge n$ and deg $U_2\ge n$ (see Figure 1.1).

We say that $G_1$ is obtained from $G$ by $n$-point splitting operation. The operation of $n$-line splitting is defined as follows.

Let $G$ be a graph and $e=UV$ be an edge of $G$ with deg $U\ge 2n-3$ with $U$ adjacent to $X_1,X_2,\dots ,X_k$, $Y_1,Y_2,\dots ,Y_h$, where $h$ and $k\ge n-2$. Let $H$ be the graph obtained from $G$ by replacing $U$ by two adjacent vertices $U_1$ and $U_2$ with $V$ adj $U_1$, $V$ adj $U_2$, $U_1$ adj $X_i$ and $U_2$ adj $Y_j$, where $1\le i\le k$ and $1\le j\le h$ and deg $U_1\ge n,$ deg $U_2\ge n$. Let $H$ be said to arise from $G$ by $n$-line splitting (see Figure 1.2). We also say that $H$ is a $n$-line splitting of $G$.

The above operations can be related to the earlier splitting operations. Let $X_{i}U=x_i\left(1\le i\le k\right)$ and $UV=e$ and $U_1{U}_2=a,U_{2}V=\gamma$. Moreover, let $X=\left\{x_1,x_2,\cdots ,x_k,e\right\}$ and $\Delta =\left\{e,a,\gamma \right\}$. We denote $H$ by $G_X^e$. If $G_X$ denote the splitting of $G$ with respect to $X$ (see [6]), then $G_X^e=G_X+a+\gamma$. If $G_X^{\prime }$ is obtained from $G$ by $n$-point splitting operation, then $G_X^e=G_X^{\prime }+\gamma$. In other words, $G_X^e\setminus \left\{a,\gamma \right\}=G_X\text{and}{G}_X^e\setminus \left\{\gamma \right\}=G_X^{\prime }$.

In the next proposition, we characterize the cycles of the graph $G_X^e$ in terms of the cycles of $G$.

In [2] Raghunathan, Shikare and Waphare generalized the splitting operation of graphs to binary matroids. Shikare, Azadi and Waphare [6, 7] extended the notions of $n$-point splitting from graphs to binary matroids. The authors in [4] determined the bases of splitting matroids and Shikare, Azanchiler and Waphare [8] characterized cocircuits of splitting matroids. We extend the $n$-line splitting from graphs to binary matroids in next Section. Also we characterize circuits of the result matroid. For the matroid theory we refer the reader to [1, 10].

Proposition 1.1. Let $G$ be a graph and $e=uv$ be an edge of $G$ with deg $u\ge 2n-3$ and $u$ adjacent to $X_1,X_2,\cdots ,X_k,Y_1,Y_2,\cdots ,Y_h$, where $h$ and $k\ge n-2$. Let $X_{i}u=x_i$ and $Y_{j}u=y_j$ for $1\le i\le k$; $1\le j\le h$. Let $X=\left\{e,x_1,x_2,\cdots ,x_k\right\}$ and $Y=\left\{y_1,y_2,\cdots ,y_h\right\}$. Then $C$ is a cycle in $G_X^e$ if and only if $C$ satisfies one of the following conditions:

• $C$ is a cycle in $G$, containing precisely two elements of $X$.
• $C$ is a cycle in $G$ containing no element of $X$.
• $C=C_1\cup C_2$, where $C_1$ and $C_2$ are edge-disjoint cycles of $G$, each contains exactly one edge of $X$ and $C_1\cup C_2$ contains, no cycle of type (1) or (2).
• $C=C_1\cup \left\{a\right\}$ when $C_1$ is a cycle of $G$, containing precisely one edge of $X$.
• $C=C_1\cup \left\{e,\gamma \right\}$, where $C_1$ is a cycle of $G$, containing exactly one element of $X-\left\{e\right\}$.
• $C=\left(C_1\setminus e\right)\cup \left\{a,\gamma \right\}$, where $C_1$ is a cycle of $G$, containing precisely one element of $X-e$ and the edge $e$.
• $C=\left(C_1\setminus e\right)\cup \left\{\gamma \right\}$, where $C_1$ is a cycle of $G$, containing the edge $e$ of $X$.
• $C=\left\{e,a,\gamma \right\}$.

2. Splitting of a Binary Matroid with Respect to an Element and a Set

Now, we extend the notion of $n$-line splitting operation from graphs to binary matroids. In the first step, we consider matrix approach to this operation in binary matroids.

Definition 2.1. Let $M$ be a binary matroid on a set $S$ and $X$ be a subset of $S,e\in X$. Suppose that $A$ is a matrix over $GF\left(2\right)$, that represents the matroid $M$. Let $A_X^e$ be the matrix that is obtained by adjoining an extra row to $A$ with this row being zero everywhere except in the columns corresponding to the elements of $X$ where it takes the value 1, and then adjoining two columns $a$ and $\gamma$ to the resulting matrix such that the column a is zero everywhere except in the last row (new row) where it takes the value 1, and $\gamma$ is a sum of two column vectors corresponding to $a$ and $e$.

Let $M_X^e$ be the vector matroid of the matrix $A_X^e$. We say that $M_X^e$ has been obtained from $M$ by splitting $e$ and $X$ in $M$. The transition from $M$ to $M_X^e$ is called splitting of $M$ with respect to $e$ and $X$. For convenience, we say that $M_X^e$ is a element -set splitting (es-splitting) matroid.

Remark 2.2. Let $r$ and $r^{\prime }$ be the rank functions of $M$ and $M_X^e$, respectively. Then $r^{\prime }\left(M_X^e\right)=r\left(M\right)+1$.

In the next proposition we characterize the circuits of the matroid $M_X^e$.

Theorem 2.3. Let $M=\left(S,\mathcal{C}\right)$ be a binary matroid, $X\subseteq S,e\in X$ and $a,\gamma \notin S$. Then $M_X^e=\left(S\cup \left\{a,\gamma \right\},{\mathcal{C}}_X^e\right)$, where ${\mathcal{C}}_X^e={\mathcal{C}}_0\cup {\mathcal{C}}_1\cup {\mathcal{C}}_2\cup {\mathcal{C}}_3\cup \left\{\Delta \right\}$ with $\Delta =\left\{e,a,\gamma \right\}$ and

${\mathcal{C}}_0=\left\{C\in \mathcal{C}\mid C\text{ contains an even number of elements of }X\right\}$;

${\mathcal{C}}_1=$ The set of minimal members of $\left\{\right.C_1\cup C_2\mid C_1,C_2\in \mathcal{C},C_1\cap C_2=\phi$ and each $C_1$ and $C_2$ contains an odd number of element of $X$ such that $C_1\cup C_2$ contains no member of ${\mathcal{C}}_0\left.\right\}$;

${\mathcal{C}}_2=\left\{C\cup \left\{a\right\}\mid C\in \mathcal{C}\text{ and }C\text{ contains an odd number of element of }X\right\}$.

${\mathcal{C}}_3=\left\{\right.C\cup \left\{e,\gamma \right\},\mid C\in \mathcal{C},e\notin C$ and $C$ contains odd number of elements of $\left.X\right\}$.

$\cup \left\{\right.\left(C\setminus e\right)\cup \left\{\gamma \right\}\mid C\in \mathcal{C},e\in C$ and $C$ contains odd number of elements of $X\left.\right\}$.

$\cup \left\{\right.\left(C\setminus e\right)\cup \left\{a,\gamma \right\}\mid C\in \mathcal{C},e\in C$ and $C\setminus e$ contains odd number of elements of $X\left.\right\}$.

Proof. We prove that ${\mathcal{C}}_X^e$ satisfies the circuit axioms of a matroid.

(1) Let $A,B\in {\mathcal{C}}_X^e$, we show that if $A\subseteq B$, then $A=B$. In other words, we prove that $A\not\subset B$ and $B\not\subset A$. The property clearly holds, if both $A$ and $B$ belong to ${\mathcal{C}}_0,{\mathcal{C}}_1$ or ${\mathcal{C}}_2$.

Now, assume that both $A$ and $B$ belong to ${\mathcal{C}}_3$ and let $A=C_1\cup \left\{e,\gamma \right\}$ and $B=\left(C_2\setminus e\right)\cup \left\{\gamma \right\}$, where $C_1,C_2\in \mathcal{C},C_1$ and $C_2$ contain an odd number of element $X,e\notin C_1$ and $e\in {\mathcal{C}}_2$. Thus $e\in A$ and $e\notin B$, so it follows that $A\not\subset B$. Similarly, $a\notin A$ and $a\in B$, implies that $B⊈A$.

Next, let $A\in {\mathcal{C}}_i$ and $B\in {\mathcal{C}}_j,i\ne j$ and $i,j=0,1,2,3$.

• Let $A\in {\mathcal{C}}_0$ and $B\in {\mathcal{C}}_1$. Then $B=C_1\cup C_2$, where $C_1$ and $C_2$ are circuits of $M$, satisfying the properties stated in the Definition 4.2.1 of ${\mathcal{C}}_1$.
• Let $A\in {\mathcal{C}}_0$ and $B\in {\mathcal{C}}_2$. Then $B=B_1\cup \left\{a\right\}$, where $B$ is a circuit of $M$, containing an odd number of element of $X$. Since $A\not\subset B_1$ and $B_1\not\subset A$, it follows that $A\not\subset B$ and $B\not\subset A$, as desired.
• Suppose $A\in {\mathcal{C}}_0$ or ${\mathcal{C}}_1$ and $B\in {\mathcal{C}}_3$. Since $\gamma \in B$ and $\gamma \notin A$, then $A\not\subset B$ and $B\not\subset A$.
• Let $A\in {\mathcal{C}}_1$ and $B\in {\mathcal{C}}_2$. Suppose $A=C_1\cup C_2$ and $B=B_1\cup \left\{a\right\}$, where $C_1,C_2$ and $B_1$ are circuits of $M$, containing an odd number of element of $X$. Since $a\in B$ and $a\notin A,B\not\subset A$. Also $A\not\subset B$, for $A\subseteq B$ implies that $A\subseteq B_1$ and $C_1\cup C_2\subset B_1$, leads to a contradiction.
• Let $A\in {\mathcal{C}}_2$ and $B\in {\mathcal{C}}_3$. Then $A=A_1\cup \left\{a\right\}$ and $B=B_1\cup \left\{e,\gamma \right\}$ or $B=\left(B_2\setminus e\right)\cup \left\{\gamma \right\}$, where $A_1,B_1$ and $B_2$ are circuits in $M$ each contains an odd number of elements of $X$ and $e\notin B_1,e\in B_2$. Since $\gamma \in B$ but $\gamma \notin A$, $B\not\subset A$. Similarly, $A\not\subset B$.

(2) Let $A,B\in {\mathcal{C}}_X^e$ and $A\ne B$. We prove that there exists $D\in {\mathcal{C}}_X^e$ such that $D\subseteq A\Delta B$.

Firstly, let $A,B\notin {\mathcal{C}}_2\cup {\mathcal{C}}_3\cup \left\{\Delta \right\}$. By binarity of $M$,

where $C_1^{\prime },C_2^{\prime },\cdots$ and $C_m^{\prime }$ are disjoint circuits of $M$. Since $A$ and $B$ each contains an even number of element of $X$, $A\Delta B$ contains an even number of element of $X$. If $A\Delta B$ contains no element of $X$, then each circuit $C_i^{\prime },i=1,2,\cdots ,m$ is a member of ${\mathcal{C}}_0$ and is contained in $A\Delta B$. If for some $j,1\le j\le m,C_j^{\prime }$ contains an even number of element of $X$, then $D=C_j^{\prime }$. Otherwise, $m$ must be an even integer and for every $j=1,2,\cdots ,m,C_j^{\prime }$ must contain an odd number of element of $X$. If $C_1^{\prime }\cup C_2^{\prime }$ contains a member of ${\mathcal{C}}_0$, say $C$, then we take $D=C$. Otherwise; $D=C_1^{\prime }\cup C_2^{\prime }$ or a minimal member of ${\mathcal{C}}_1$ contained in it.

Secondly, suppose $A,B\in {\mathcal{C}}_2\cup {\mathcal{C}}_3\cup \left\{\Delta \right\}$, then we have the following cases:

(I) Let $A,B\in {\mathcal{C}}_2$ and $A=A_1\cup \left\{a\right\},B=B_1\cup \left\{a\right\}$. Then $A\Delta B=A_1\Delta B_1=C_1^{\prime }\cup C_2^{\prime }\cup \cdots \cup C_m^{\prime }$, where $C_1^{\prime },C_2^{\prime },\cdots$ and $C_m^{\prime }$ are disjoint circuits of $M$. Since each of $A$ and $B$ contains an odd number of elements of $X,A\Delta B$ contains an even number of elements of $X$. By similar argument as above, we can find $D\in {\mathcal{C}}_X^e$ such that $D\subseteq A\Delta B$.

(II) Let $A$ and $B$ belong to ${\mathcal{C}}_3$, we have the following subcases:

(i) $A=A_1\cup \left\{e,\gamma \right\}$ and $B=B_1\cup \left\{e,\gamma \right\}$. (ii) $A=\left(A_1\setminus e\right)\cup \left\{\gamma \right\}$ and $B=\left(B_1\setminus e\right)\cup \left\{\gamma \right\}$. (iii) $A=A_1\cup \left\{e,\gamma \right\}$, $B=\left(B_1\setminus e\right)\cup \left\{\gamma \right\}$. (iv) $A=A_1\cup \left\{e,\gamma \right\}$, $B=\left(B_1\setminus e\right)\cup \left\{a,\gamma \right\}$. (v) $A=\left(A_1\setminus e\right)\cup \left\{a,\gamma \right\}$ and $B=\left(B_1\setminus e\right)\cup \left\{a,\gamma \right\}$. (vi) $A=\left(A_1\setminus e\right)\cup \left\{a,\gamma \right\}$, $B=\left(B_1\setminus e\right)\cup \left\{a,\gamma \right\}$.

In cases (i), (ii) and (vi), we have $A\Delta B=A_1\Delta B_1=C_1^{\prime }\cup C_2^{\prime }\cup \cdots \cup C_m^{\prime }$, where $C_1^{\prime },C_2^{\prime },\cdots ,C_m^{\prime }$ are disjoint circuits of $M$. By the similar arguments as in (1), we can find $D\in {\mathcal{C}}_X^e$ such that $D\subseteq A\Delta B$. In case (iii), we have $A\Delta B=\left(A_1\Delta B_1\right)\cup \left\{a\right\}$.

Let $A_1\Delta B_1=C_1^{\prime }\cup C_2^{\prime \prime }\cup \cdots \cup C_n^{\prime \prime }$, where $C_1^{\prime \prime },C_2^{\prime \prime },\cdots ,C_n^{\prime \prime }$ are disjoint circuits of $M$. If $C_1^{\prime \prime }$ contains an even number of elements of $X$, then it will be an element of ${\mathcal{C}}_X^e$ which is contained in $A_1\Delta B_1$, and hence in $A\Delta B$. If $C_1^{\prime \prime }$ contains an odd number of elements of $X$, then $C_1^{\prime \prime }\cup \left\{a\right\}$ is an element of ${\mathcal{C}}_2$, contained in $A\Delta B$. By similar argument, the cases (iv) and (v) follow.

(III) Let $A\in {\mathcal{C}}_2$ and $B\in {\mathcal{C}}_3$. Then $A=A_1\cup \left\{a\right\}$ and for $B$, we have two subcases: (1) $B=B_1\cup \left\{e,\gamma \right\}$, and (2) $B=\left(B_1\setminus e\right)\cup \left\{\gamma \right\}$. Thus, in (1), we have $A\Delta B=\left(A_1\Delta B_1\right)\cup \left\{e,a,\gamma \right\}$, where $A_1$ and $B_1$ are circuits of $M$, containing an odd number of element of $X$. Clearly, $A_1\Delta B_1\subseteq A\Delta B$. Let $A_1\Delta B_1=C_1^{\prime }\cup C_2^{\prime }\cup C_m^{\prime }$, where $C_1^{\prime },C_2^{\prime },\cdots ,C_m^{\prime }$ are disjoint circuits of $M$. By similar arguments as in (2), we can find $D\in {\mathcal{C}}_X^e$ such that $D\subseteq A_1\Delta B_1$ and hence $D\subseteq A\Delta B$. In (2), we have $A=A_1\cup \left\{a\right\}$ and $B=\left(B_1\setminus e\right)\cup \left\{\gamma \right\}$. So $A\Delta B=\left[\left(A_1\Delta B_1\right)\setminus e\right]\cup \left\{a,\gamma \right\}$, where $A_1$ and $B_1$ are circuits in $M$. By the argument as given above, we can find $D\in {\mathcal{C}}_X^e$, such that $D\subseteq A\Delta B$.

(IV) Let $A=\Delta$ and $B\in {\mathcal{C}}_2$. Then $B=B_1\cup \left\{a\right\}$, where $B_1$ is a circuit in $M$ containing an odd number of element of $X$. We have two subcases: (i) $e\notin B_1$. Then $A\Delta B$ is a circuit of $M$, containing an even number of elements of $X$. Thus $D=A\Delta B$. (ii) $e\in B_1$. Then $A\Delta B$ is also a circuit in $M_X^e$ and so $D=A\Delta B$.

(V) Let $A=\Delta$ and $B\in {\mathcal{C}}_3$. Then $B=B_1\cup \left\{e,\gamma \right\}$ or $B=\left(B_1\setminus e\right)\cup \left\{\gamma \right\}$, where $B_1$ is a circuit of $M$, containing an odd number of element of $X$. Since $A\Delta B_1\subset A\Delta B,$ we can find $D=\Delta$, where $D\in {\mathcal{C}}_X^e$ such that $D\subseteq A\Delta B_1$, and hence $D\subseteq A\Delta B$.

We conclude that ${\mathcal{C}}_X^e$ is a collection of circuits of a binary matroid on the set $S\cup \left\{a,\gamma \right\}$.

$\square$

Definition 2.4. With notation as above, we call the matroid $\left(S\cup \left\{a,\gamma \right\},{\mathcal{C}}_X^e\right)$ as the es-splitting of $M=\left(S,\mathcal{C}\right)$ and denote it by $M_X^e$. Thus $M_X^e=\left(S^{\prime },{\mathcal{C}}_X^e\right)$, where $S^{\prime }=S\cup \left\{a,\gamma \right\}$.

Example 2.5. Consider the matroid $M=F_7$ with ground set $S=\left\{1,2,3,4,5,6,7\right\}$ and the set of circuits

Let $X=\left\{1,2,4\right\}$ and $e=1$. Then the circuit set of $M_X^e$ is

Theorem 2.6. The matrix $A_X^e$ represents the splitting matroid $M_X^e$.

3. Connectedness of the Splitting Matroid $M_X^e$

In [5] Shikare characterized the connectedness in splitting matroid$M_X$. The next theorem characterize connectedness of $M_X^e$.

Theorem 3.1. Let $M=\left(S,\mathcal{C}\right)$ be a binary connected matroid. Then $M_X^e$ is connected.

Proof. Let $M$ be a connected matroid on $S$. Then for every pair $x,y\in S$ there is a circuit of $M$ containing $x$ and $y$. We show that for any two elements $x$ and $y$ belonging to $S\cup \left\{a,\gamma \right\}$, there is a circuit of $M_X^e$ containing $x$ and $y$. We consider the following cases:

• Let $x,y\in \left\{a,\gamma \right\}$. Then $x,y\in \Delta$ and we are through.
• Suppose $x,y\notin \left\{a,\gamma \right\}$. Then $x,y\in S$ and $M$ has a circuit, say $C$ containing $x$ and $y$. We have the following two subcases:

  (i) $C$ contains an even number of elements of $X$. Then we are through.

  (ii) $C$ contains an odd number of elements of $X$. Then $C\cup \left\{a\right\}$ is a circuit of $M_X^e$ containing $x$ and $y$.

• Let $x=a$ and $y\in S$. Then there is a circuit $C\in \mathcal{C}$ such that $y,e\in C$. We have two subcases:

  (i) Suppose $C$ contains an even number of elements of $X$. Then $C\in {\mathcal{C}}_X^e,e\in C\cap \Delta ,y\in C$ and $a\in \Delta$. By [3], there is a circuit say $C^{\prime }$ of $M_X^e$ such that $a,y\in C^{\prime }$.

  (ii) Let $C$ contains an odd number of elements of $X$. Then $C\cup \left\{a\right\}$ is a circuit in $M_X^e$ and $a,y\in C\cup \left\{a\right\}$.

• Let $x=\gamma$ and $y\in S$. Then there is a circuit $C\in \mathcal{C}$ such that $y,e\in C$. We consider the following two subcases:

  (i) $C$ contains an even number of element of $X$. Then $C\in {\mathcal{C}}_X^e,e\in C\cap \Delta ,y\in C$ and $\gamma \in \Delta$, so by [3], there is a circuit of $M_X^e$ containing $\gamma$ and $y$.

  (ii) $C$ contains an odd number of element of $X$. Then $\left(C\setminus e\right)\cup \left\{\gamma \right\}$ is a circuit of $M_X^e$ containing $\gamma$ and $y$. $\square$

Remark 3.2. Converse of Theorem 3.1 is not true. For example, let $M$ be a cycle matroid of the graph $G$ (See Figure 3.1). Let $X=\left\{e,x_1,x_2\right\}$. Then $M_X^e$ is a cycle matroid of $G_X^e$.

We observe that the matroid $M=M\left(G\right)$ is disconnected while $M_X^e=M\left(G_X^e\right)$ is connected.

The next theorem shows that one can obtain a connected matroid from disconnected matroid with the help of es-splitting operation.

Theorem 3.3. Let $M$ be a bridgeless binary matroid on $S$ with $n$ components $M_1,M_2,\cdots ,M_n$. Let $x_i$ be chosen from $M_i$ for $i=1,2,\cdots ,n$ and $X=\left\{x_1,x_2,\cdots ,x_n\right\},e\in X$. Then $M_X^e$ is a connected matroid on $S\cup \left\{a,\gamma \right\}$.

Proof. Let $M_1=\left(S_1,D_1\right),M_2=\left(S_2,D\right),\cdots ,M_n=\left(S_n,D_n\right)$ be the $n$ components of $M$, where $D_i$ denote the collection of circuits of $M_i$. Then $S_i\cap S_j=\phi$ for $i\ne j,i,j=1,2,\cdots ,n$ and ${\cup }_{i=1}^n{S}_i=S$. $X=\left\{x_1,x_2,\cdots ,x_n\right\}$, where $x_i\in S_i$ for $i=1,2,\cdots ,n$. Since $e\in X$, there is a $j$ such that $x_j=e$, where $1\le j\le n$. Suppose ${\mathcal{A}}_i=\left\{C\in D_i\mid x_i\notin C\right\}$ for $i=1,2,\cdots ,n$ and ${\mathcal{A}}_{n+1}=\left\{\right.C_i\cup C_j\mid C_i\in D_i,C_j\in D_j,x_i\in C_i,x_j\in C_j,i\ne j$ and $i,j=1,2,\cdots ,n\left.\right\}$. Let ${\mathcal{ℬ}}_i=\left\{C\cup \left\{a\right\}\mid C\in D_i,x_i\in C,i=1,2,\cdots ,n\right\}$ and

The matroid $M_X^e$ has the circuit set ${\mathcal{C}}_X^e$, where ${\mathcal{C}}_X^e={\mathcal{A}}_1\cup {\mathcal{A}}_2\cup \cdots \cup {\mathcal{A}}_n\cup {\mathcal{A}}_{n+1}\cup B_1\cup B_2\cup \cdots \cup B_n\cup {\mathcal{C}}_1\cup {\mathcal{C}}_2\cup \cdots \cup {\mathcal{C}}_n\cup \left\{\Delta \right\}$.

Claim. For every pair of elements $\alpha ,\beta$ of $S\cup \left\{a,\gamma \right\}$ there is a circuit of $M_X^e$, containing $\alpha$ and $\beta$. We have the following cases:

• Let $\alpha ,\underset{̱}{\in }S$. Then we consider the following subcases:

  (i) Suppose $\alpha$ and $\beta$ belong to one component, say $M_i$. Then there is a circuit say $C_i$ of $M_i$, containing $\alpha$ and $\beta$. If $x_i$ does not belong to $C_i$, then $C_i$ is a circuit of $M_X^e$, containing $\alpha$ and $\beta$. If $x_i\in C_i$, then for any $j\ne i$, $C_j$ is a circuit of $M_j$ containing $x_j$. Thus $C_i\cup C_j$ is a circuit of $M_X^e$ containing $\alpha$ and $\beta$.

  (ii) Let $\alpha$ and $\beta$ belongs to different components of $M$, say $\alpha \in M_i$ and $\underset{̱}{\in }{M}_j$. Consider the circuits $C_i$ and $C_j$, where $C_i$ contains $\alpha$ and $x_i$ and $C_j$ contains $\beta$ and $x_j$. Then $C_i\cup C_j$ is a required circuit of $M_X^e$.

• Let $\alpha \in S$ and $\underset{̱}{\in }\left\{a,\gamma \right\}$. Then $\alpha \in S_i$ for some $i$. Consequently, there is a circuit of $M_i$, say $C_i$, containing $\alpha$ and $x_i$. If $\underset{̱}{=}a$ then $C_i\cup \left\{a\right\}$ is a circuit of $M_X^e$ containing $\alpha$ and $\beta$. If $b=\gamma$, then $\left(C_i\setminus \left\{e\right\}\right)\cup \left\{\gamma \right\}$ is a circuit of $M_X^e$ containing $\alpha$ and $\beta$.
• If $\alpha =a$ and $\underset{̱}{=}\gamma$, then $\alpha ,\underset{̱}{\in }\Delta$, a 3-circuit in $M_X^e$. $\square$

References

Department of Mathematics, University of Urmia, Urima–57135 (Iran)

E-mail address: azanchiler@yahoo.com

Received August 12, 2006