Abstract
We obtain some results concerning the planarity and graphicness of the splitting matroids. Further, we explore the effect of splitting operation on the sum of two matroids.
1. Introduction
The matroid notations and terminology used here will follow Oxley [1].
Fleischner [2] defined the splitting operation for a graph with respect to a pair of adjacent edges as follows.
Let be a connected graph and let be a vertex of degree at least three in . If and are two edges incident at , then splitting away the pair from results in a new graph obtained from by deleting the edges and , and adding a new vertex adjacent to and . The transition from to is called the splitting operation on . Figure 1 illustrates this construction explicitly.
Fleischner [2] used the splitting operation to characterize Eulerian graphs. Fleischner [3] also developed an algorithm to find all distinct Eulerian trails in an Eulerian graph using the splitting operation. Tutte [4] characterized 3-connected graphs, and Slater [5] classified 4-connected graphs using a slight variation of this operation.
Raghunathan et al. [6] extended the splitting operation from graphs to binary matroids as follows.
Definition 1.1. Let be a binary matroid and suppose . Let be the matrix obtained from by adjoining the row that is zero everywhere except for the entries of in the columns labeled by and . The splitting matroid is defined to be the vector matroid of the matrix .
Alternatively, the splitting operation can be defined in terms of circuits of binary matroids as follows.
Lemma 1.2 (see [6]). Let be a binary matroid on a set together with the set of circuits and let . Then with , where or ; and and contains no member of .
Shikare and Waphare [7] characterized graphic matroids whose splitting matroids are also graphic. In fact, they proved the following theorem.
Theorem 1.3 (see [7]). The splitting operation, by any pair of elements, on a graphic matroid yields a graphic matroid if and only if the matroid has no minor isomorphic to the cycle matroid of any of the four graphs of Figure 2.
We define a matroid to be planar if it is both graphic and cographic. Let us consider the graph (i) of Figure 2. In this graph by splitting away two non-adjacent elements and , we will get a matroid which is not planar. This example exhibits the fact that even if the original matroid is planar, there exist pairs of non-adjacent elements and such that the splitting matroid is not planar.
Now, by considering above example, we prove the following Theorem.
Theorem 1.4. Let be a planar graph. Then there is at least one pair of non-adjacent edges and such that is a planar graph.
The proof of the theorem is given in Section 3.
Theorem 1.5 (see [1]). A binary matroid is graphic if and only if it has no minor isomorphic to , or .
The following example exhibits the fact that there exists a graphic matroid and a pair of non-adjacent edges, such that the splitting of with respect to this pair does not yield a graphic matroid.
Example 1.6. Consider the matroid where is the complete graph on vertices as shown in Figure 3. The matroid has the ground set and the collection of circuits
The matroid is graphic and it arises as the cycle matroid of the graph of Figure 3.
Consider the non-adjacent elements and of . Then, by Lemma 1.2, is the matroid with ground set and circuit set . By contracting in , we get a matroid , which is isomorphic to the matroid . Thus, by Theorem 1.5, is not graphic. By similar arguments, we can check that splitting of by any other pair of non-adjacent edges is not graphic. With this observation, we state the following theorem for the splitting matroids.
Theorem 1.7. Let be a graphic matroid that is not isomorphic to . Then there exists at least one pair of non-adjacent edges and of such that is graphic.
The proof of the theorem is given in Section 3.
Shikare et al. [8] introduced the concept of generalized splitting operation for a graph with respect to adjacent edges in the following way.
Definition 1.8. Let be a connected graph and a vertex of with . Let be a set of adjacent edges incident at . Then splitting away the edges in from results in a new graph obtained from by deleting the edges and adding a new vertex adjacent to . We say that the graph has been obtained from by splitting away the edges or in short with respect to the set .
This construction is illustrated in Figure 4 where .
Shikare et al. [8] later on extended the notion of the generalized splitting operation from graphs to binary matroids in the following way.
Definition 1.9. Let be a binary matroid on a set and a subset of . Suppose that is a matrix over that represents the matroid . Let be the matrix that is obtained by adjoining an extra row to with this row being zero everywhere except in the columns corresponding to the elements of where they take the value . Let be the matroid represented by the matrix . We say that has been obtained from by splitting the set . The transition from to is called a generalized splitting operation.
In this paper, we explore the effect of the splitting operation on the sum of two matroids and give some application of these results.
2. The Splitting Operation on the Sum of Two Matroids
In this section, we provide some definitions and the results which are used in the proof of theorems.
Proposition 2.1 (see [1]). The following statements are equivalent for a graph :(i)is a planar graph;(ii) is a planar matroid;(iii) has no minor isomorphic to or .
Proposition 2.2 (see [1]). Let and be disjoint subsets of . Then(i);(ii);(iii).
Proposition 2.3 (see [7]). Let and be two elements of a binary matroid . Then (i) if and only if and are in series or both and are coloops in ;(ii).
Definition 2.4 (see [9]). If and are two (not necessarily different) matroids on the same set then let us define , where . The matroid is called the sum of two matroids.
Theorem 2.5 (see [9, 10]). Suppose are matroids on the same set . If the real matrices coordinates the respective matroids and all the nonzero entries of all these matrices are algebraically independent over the field of the rationales, then the matrix coordinates the matroid .
Theorem 2.6 (see [6]). Suppose is a binary matroid on a set and . Then is Eulerian if and only if is Eulerian.
Definition 2.7 (see [1]). Let and be matroids on disjoint sets and . Let and . Then is a matroid. This matroid is called the direct sum or 1-sum of and and is denoted .
In the next theorem, Shikare et al. [8] explored the relation between direct sums of two matroids and the splitting operation.
Theorem 2.8 (see [8]). Let and be two binary matroids with . Let and be subsets of and , respectively. If there is no circuit of or containing an odd number of elements of and , respectively, then .
3. Applications
In this section, we use definitions and the results of Section 2 to prove Theorems 1.4 and 1.7. We also explore the effect of the splitting operation on the sum of two matroids.
Proof of Theorem 1.4. Suppose is a planar graph. By Proposition 2.1, is a planar matroid, and therefore has no minor isomorphic to or . We claim that is planar for at least one non-adjacent pair of edges and of .
On the contrary, suppose that is not planar for every non-adjacent pair of edges and of . This implies that has a minor isomorphic to or . First, suppose that has as a minor. Since no two elements in are in series, and cannot be elements of . Let and . We have . This implies that . By Proposition 2.2, we have . Now, by Proposition 2.3 (ii), we have . So . We conclude that has a minor isomorphic to which is a contradiction to our assumption that is a planar matroid. So cannot have as a minor. Now, suppose that has as a minor. Then, by similar arguments, we arrive to a contradiction. We conclude that cannot have as a minor as well. Therefore, is planar for at least one pair of non-adjacent edges and . Consequently, is planar for at least one pair of non-adjacent edges and .
Now, we use Theorem 1.3 to prove Theorem 1.7.
Proof of Theorem 1.7. Let be a graphic matroid. Then is isomorphic to a cycle matroid of some graph . Suppose is not isomorphic to . By Theorem 1.3, the splitting operation, by any pair of elements, on a graphic matroid yields a graphic matroid if and only if the matroid has no minor isomorphic to the cycle matroid of any of the four graphs of Figure 2. If has no minor isomorphic to the cycle matroid of any of the four graphs of Figure 2, then there exists at least one pair of non-adjacent edges and of such that is graphic. If has minor isomorphic to the cycle matroid of any of the three graphs (i), (ii), and (iii) of Figure 5, then each of these graphs contains at least one pair of non-adjacent edges and of such that is graphic.
As proved in Example 1.6 in Section 1, if , there is no pair of non-adjacent edges and such that is graphic and by Theorem 1.3, is only matroid which has this property and this completes the proof.
In the next theorem, we explore the relation between the sum of two matroids and the splitting operation.
Theorem 3.1. Let and be binary matroids on the same underlying set . If , then .
Proof. Let and be the matrices that represent and , respectively. The matrix representation for and , say and , is obtained from and , respectively, by adjoining extra rows to and , which are zero everywhere except in the columns corresponding to the elements and where it takes the value .
By Theorem 2.5, the matrix representation for is the matrix with two equal rows. By replacing a row by the sum of that row and the row which is equal to that, we get one zero row. By deleting the zero row, we obtain a matrix representation of the matroid .
The matrix representation of , say , can be obtained by adjoining one extra row to the matrix which is zero everywhere except in the columns corresponding to the elements and where it takes the value . We observe that . This completes the proof.
It is well known that a graph is Eulerian if and only if its edge set can be partitioned into disjoint circuits. Generalizing this graph theoretic concepts, Welsh [11] defined Eulerian matroid. A matroid is said to be Eulerian if the ground set is the union of disjoint circuits of the matroid. Further, Welsh [11] proved that a binary matroid is Eulerian if and only if its dual matroid is bipartite.
The following theorem states that the splitting of the sum of two binary Eulerian matroids is the sum of the corresponding splitting matroids.
Theorem 3.2. Let and be binary matroids with . Then and are Eulerian if and only if is Eulerian.
Proof. Let and be the matrices that represent and , respectively. Suppose and are Eulerian matroids. By Theorem 2.6, and are Eulerian. Let and be the matrices obtained from and by adjoining extra rows, which are zero everywhere except in the columns corresponding to the elements and where they take the value . Then the matrix is a matrix representation of the matroid . Since the number of 1’s in each row of the matrix is even, the matroid is Eulerian.
The converse part of proof is straightforward.
Using Theorem 3.2, we prove that the splitting of the sum of binary Eulerian matroids is the sum of the corresponding splitting matroids.
Theorem 3.3. Let , ,…, be binary matroids with . Then are Eulerian if and only if is Eulerian.
Proof. Let be the matrices that represent , respectively. Suppose that are Eulerian matroids. By Theorem 2.6, are Eulerian. Let be the matrices obtained from by adjoining extra rows, which are zero everywhere except in the columns corresponding to the elements and where they take the value . Then the matrix is a matrix representation of the matroid . Since the number of 1’s in each row of the matrix is even, the matroid is Eulerian.
The converse part of proof is straightforward.
Theorem 3.4. Let and be two binary matroids with . Let and be subsets of and , respectively. Then and are Eulerian if and only if is Eulerian.
Proof. Let and be two binary matroids with . Let and be subsets of and , respectively. Suppose the and are Eulerian. Then, and can be expressed as a disjoint unions of circuits of and , respectively. Suppose that and . Then is a disjoint union of circuits of . So is Eulerian. By Theorem 2.8, is Eulerian.
The converse part of the proof is straightforward.