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ISRN Discrete Mathematics
Volume 2012 (2012), Article ID 850538, 7 pages
http://dx.doi.org/10.5402/2012/850538
Research Article

Splitting Lemma for 2-Connected Graphs

Department of Mathematics, University of Pune, Pune 411007, India

Received 1 October 2012; Accepted 17 October 2012

Academic Editors: S. Bozapalidis, E. Kranakis, and W. Wang

Copyright © 2012 Y. M. Borse. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Using a splitting operation and a splitting lemma for connected graphs, Fleischner characterized connected Eulerian graphs. In this paper, we obtain a splitting lemma for 2-connected graphs and characterize 2-connected Eulerian graphs. As a consequence, we characterize connected graphic Eulerian matroids.

1. Introduction

Fleischner [1] introduced a splitting operation to characterize Eulerian graphs as follows. Let be a connected graph and with . If and are two edges incident with , then splitting away the pair of edges from the vertex results in a new graph obtained from by deleting the edges and , and adding a new vertex adjacent to and (see Figure 1).

850538.fig.001
Figure 1

The following splitting lemma established by Fleischner [1] has been widely recognized as a useful tool in the graph theory.

Splitting Lemma (see [1, page III-29]). Let be a connected bridgeless graph. Suppose such that and are the edges incident with . Form the graphs and by splitting away the pairs and , respectively, and assume and belong to different blocks if is a cut vertex of . Then either or is connected and bridgeless.

This lemma is used to obtain the following characterization of Eulerian graphs.

Theorem 1.2 (see [1, page V-6]). A graph has an Eulerian trail if and only if can be transformed into a cycle through repeated applications of the splitting procedure on vertices of a degree exceeding . Moreover, the number of Eulerian trails of equals the number of different labeled cycles into which can be transformed this way.

Thus a connected graph is Eulerian if and only if there exists a sequence of connected graphs such that is a cycle and is obtained from by applying splitting operation once.

The splitting operation may not preserve -connectedness of the graph. Consider the graph of Figure 2. It is -connected but the graph is not -connected for any two adjacent edges and .

850538.fig.002
Figure 2

We obtain the splitting lemma for -connected graphs as follows.

Theorem 1.3. Let be a -connected graph and let be a vertex of with . Then either is -connected for some pair of edges incident with or for any pair of edges incident with ; there is another pair of adjacent edges of such that is -connected.

The next theorem is a consequence of the above result.

Theorem 1.4. Let be a -connected graph. Then is Eulerian if and only if there exists a sequence of -connected graphs such that is a cycle and is obtained from by applying splitting operation once or twice for .

A matroid is Eulerian if its ground set can be partitioned into disjoint circuits, and it is connected if any pair of its elements is contained in a circuit. It is clear that an Eulerian matroid may not be connected. A matroid is graphic if it is isomorphic to the cycle matroid of a graph. For matroid concepts and terminology, we refer to Oxley [2]. Raghunathan et al. [3] generalized the splitting operation of graphs to binary matroids and characterized Eulerian matroids in terms of this operation. We characterize connected Eulerian graphic matroids.

In Section 2, we prove Theorems 1.3 and 1.4. The matroid extension is considered in Section 3.

2. Eulerian 2-Connected Graphs

A block of a connected graph is a pendant block if it contains exactly one cut vertex of . For an edge , we denote the set of end vertices of by . For a vertex of , let denote the set of edges of which are incident with , that is, . Raghunathan et al. [3] characterized the circuits of the graph in terms of circuits of as follows.

Lemma 2.1 (see [3]). Let be a graph and let be a pair of adjacent edges of . Then a subset of edges of the graph is a circuit in if and only if satisfies one of the following conditions:(i) is a circuit in containing and ;(ii) is a circuit in containing neither nor ;(iii) , where and are edge disjoint circuits of with , , and does not contain a circuit in satisfying either or above.

Lemma 2.2. Let be a -connected graph and be a vertex of with and such that the graph is not -connected. Then is connected and has exactly two pendant blocks. Further, one pendant block contains , and the other pendant block contains .

Proof. The proof is straightforward (see Figure 3).

850538.fig.003
Figure 3

Lemma 2.3. Let be a -connected graph and let be a vertex of with such that is not -connected for all . Then, for a given , the graph is connected and has one cut vertex and two blocks.

Proof. Let be a pair of edges incident with . By Lemma 2.2, is connected and has exactly two pendant blocks, say and . We may assume that contains and contains . As , we can choose two edges , from . Let and be paths in from to with , , and . Each of and corresponds to a cycle in . By Lemma 2.1, these cycles are preserved in the graph . Therefore is contained in a block of for . By Lemma 2.2, has two pendant blocks one containing edges , and the other containing . Hence and belong to different pendant blocks of . Hence and share at most one vertex of . However, and share all cut vertices of . This implies that has exactly one cut vertex. Therefore, by Lemma 2.2, is connected and has exactly two blocks.

Lemma 2.4. Let and be as stated in Lemma 2.3. Then there exists a vertex in such that is the cut vertex of for all .

Proof. Let . By Lemma 2.3, is connected and has one cut vertex, say . Let . Then, by Lemma 2.3, is also connected and has two blocks and one cut vertex. It suffices to prove that is the cut vertex of . If , then there is nothing to prove. Suppose . We may assume that and . By Lemma 2.3, is connected and has two blocks, say and . By Lemma 2.2, we may assume that contains and contains . Let be an edge of incident with such that . Let and be paths in from the vertex to such that , , and , . Then each of and contains all cut vertices of . Therefore is a common vertex of and . Further, each of and corresponds to a cycle in . By Lemma 2.1, these cycles are preserved in the graph and hence are contained in blocks of . Therefore contains for . Thus is a common vertex of and . This implies that is a cut vertex of . By Lemma 2.3, is the only cut vertex of . Let and be paths in from to such that and . Then contains the cut vertex and, further, it corresponds to a cycle in for . By Lemma 2.1, corresponds to a cycle of the graph and hence is contained in a block of for . These cycles are contained in different blocks of . By Lemma 2.2, one block of contains the edges , and the other block contains the remaining edges of that are incident with . Hence and belong to different blocks of . As is a common vertex of and , it is a cut vertex of . Thus is the cut vertex of .

Lemma 2.5. Let be a -connected graph and be a vertex of with the set of neighbours , where . Suppose is not -connected for all . Then there exists a vertex in such that for any -path and -path in with .

Proof. By Lemma 2.4, there exists a vertex in such that it is the cut vertex of for all . Let be a -path and be a -path in with . We prove that . If or is a trivial graph, then there is nothing to prove. Assume that and . Without loss of generality, we may assume that and . Let and . Then is connected and has two blocks, say and (see Figure 4). By Lemma 2.2, we may assume that contains and contains . Since is the cut vertex of , the paths are contained in . Let and . Let be a -path in containing the edge but avoiding . Let be an -path in containing and avoiding . Then for . Let and . Then each of and corresponds to a cycle in . Further, contains , and contains , . Therefore, by Lemma 2.1, and correspond to cycles in . By Lemmas 2.2 and 2.3, has exactly two blocks one of them contains and the other contain . Hence and can share at most one vertex. This implies that and can share at most one vertex. Thus .

850538.fig.004
Figure 4

Proof of Theorem 1.3. Let be a -connected graph and let be a vertex of with . Suppose is not -connected for every pair of edges incident with . Let be the set of neighbours of . Let and be any two edges of incident with . We may assume that and . By Lemma 2.5, there exists a vertex in such that for any with , where is a -path and is a -path in (see Figure 5). It is easy to see that . If for some , then is the trivial graph containing only the vertex . If , then set . If , then set . In other cases, and are nontrivial graphs and hence we can take and . It is easy to see that is -connected.

850538.fig.005
Figure 5

Now, we prove Theorem 1.4. Let denotes the degree of a vertex in a graph .

Proof of Theorem 1.4. Let be an Eulerian -connected graph. Suppose is not a cycle. Then has a vertex of a degree of at least 4. By Theorem 1.3, we get a pair of edges incident with such that either is -connected or is 2-connected for some pair of edges of having a common vertex other than . Denote this new 2-connected graph by . If , then and for any . If , then , , and for any , where is the common vertex of and other than . Further, the new vertices of that are created in the splitting procedure have degree two. Obviously, is Eulerian. If is not a cycle, then we obtain a 2-connected Eulerian graph from by applying splitting operation once (or twice) which results in reducing the degree of a vertex (or two vertices) of by 2. By repeating the same procedure and through a sequence of once or twice splitting operations performed in such a way that at each step the resulting graph is still -connected one finally arrives at a cycle which corresponds to an Eulerian trail of . The converse is obvious.

3. Eulerian 2-Connected Matroids

In this section, we extend Theorem 1.4 to connected Eulerian matroids. Raghunathan et al. [3] generalized the splitting operation of graphs to binary matroids and characterized Eulerian matroids in terms of this operation. In this section, we characterize connected Eulerian graphic matroids.

Definition 3.1 (see [3]). 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 . The transition from to is called a splitting operation. The splitting operation for binary matroids is also studied in [36].

We need the following three results.

Lemma 3.2 (see [3]). If denotes the circuit matroid of a graph , then for a pair of adjacent edges in a graph .

Lemma 3.3 (see [3]). Let be a binary matroid and . Then is Eulerian if and only if is Eulerian.

Theorem 3.4 (see [2, page 127). Let be a loopless graph without isolated vertices. If has at least three vertices, then is a connected matroid if and only if is a -connected graph.

We obtain the following characterization of connected Eulerian graphic matroids.

Theorem 3.5. Let be a connected graphic matroid. Then is Eulerian if and only if it can be transformed into a circuit through a sequence of connected graphic matroids such that is obtained from by applying splitting operation once or twice.

Proof. Let be a connected graphic matroid. Then is isomorphic to a cycle matroid of some graph . In view of Theorem 3.4, we may assume that is -connected. Suppose is Eulerian. Then the graph is Eulerian. By Theorem 1.4, there is a sequence of -connected graphs such that is a cycle, and is obtained from by applying splitting operation once or twice for . Let for . By Theorem 3.4, each is connected. It follows from Lemma 3.2 that if is obtained from by applying splitting operation once or twice then is obtained from by applying splitting operation once or twice, respectively. Further, by Lemma 3.3, is Eulerian for .
Conversely, suppose there exists a sequence of connected graphic matroids , where is obtained from by applying splitting operation once or twice for . Since is Eulerian, by Lemma 3.3, is Eulerian. By repeated applications of Lemma 3.3, we see that is Eulerian for each . Thus is Eulerian.

Acknowledgment

The author would like to thank the referees for their valuable suggestions. The paper is supported by University of Pune under BCUD project scheme.

References

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