Splitting Lemma for 2-Connected Graphs

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 𝐺 be a connected graph and 𝑣∈𝑉(𝐺) with 𝑑(𝑣)≥3. If 𝑥=𝑣𝑣₁ and 𝑦=𝑣𝑣₂ are two edges incident with 𝑣, then splitting away the pair {𝑥,𝑦} of edges from the vertex 𝑣 results in a new graph 𝐺_𝑥,𝑦 obtained from 𝐺 by deleting the edges 𝑥 and 𝑦, and adding a new vertex 𝑣_𝑥,𝑦 adjacent to 𝑣₁ and 𝑣₂ (see Figure 1).

Figure 1 

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

Splitting Lemma 𝟏.𝟏 (see [1, page III-29]). Let 𝐺 be a connected bridgeless graph. Suppose 𝑣∈𝑉(𝐺) such that 𝑑(𝑣)≥3 and 𝑥,𝑦,𝑧 are the edges incident with 𝑣. Form the graphs 𝐺_𝑥,𝑦 and 𝐺_𝑥,𝑧 by splitting away the pairs {𝑥,𝑦} and {𝑥,𝑧}, respectively, and assume 𝑥 and 𝑧 belong to different blocks if 𝑣 is a cut vertex of 𝐺. Then either 𝐺_𝑥,𝑦 or 𝐺_𝑥,𝑧 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 𝐺 has an Eulerian trail 𝑇 if and only if 𝐺 can be transformed into a cycle 𝐶 through repeated applications of the splitting procedure on vertices of a degree exceeding 2. Moreover, the number of Eulerian trails of 𝐺 equals the number of different labeled cycles into which 𝐺 can be transformed this way.

Thus a connected graph 𝐺 is Eulerian if and only if there exists a sequence 𝐺=𝐺₀<𝐺₁<⋯<𝐺_𝑛 of connected graphs such that 𝐺_𝑛 is a cycle and 𝐺_𝑖+1 is obtained from 𝐺_𝑖 by applying splitting operation once.

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

Figure 2 

We obtain the splitting lemma for 2-connected graphs as follows.

Theorem 1.3. Let 𝐺 be a 2-connected graph and let 𝑣 be a vertex of 𝐺 with 𝑑(𝑣)≥4. Then either 𝐺_𝑥,𝑦 is 2-connected for some pair {𝑥,𝑦} of edges incident with 𝑣 or for any pair {𝑥₁,𝑦₁} of edges incident with 𝑣; there is another pair {𝑥₂,𝑦₂} of adjacent edges of 𝐺 such that (𝐺_𝑥1,𝑦1)_𝑥2,𝑦2 is 2-connected.

The next theorem is a consequence of the above result.

Theorem 1.4. Let 𝐺 be a 2-connected graph. Then 𝐺 is Eulerian if and only if there exists a sequence of 2-connected graphs 𝐺=𝐺₀<𝐺₁<𝐺₂<⋯<𝐺_𝑛 such that 𝐺_𝑛 is a cycle and 𝐺_𝑖+1 is obtained from 𝐺_𝑖 by applying splitting operation once or twice for 𝑖=0,1,2,…,𝑛−1.

A matroid 𝑀 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 𝐺 is a pendant block if it contains exactly one cut vertex of 𝐺. For an edge 𝑒∈𝐸(𝐺), we denote the set of end vertices of 𝑒 by 𝑉(𝑒). For a vertex 𝑣 of 𝐺, let 𝐸_𝐺(𝑣) denote the set of edges of 𝐺 which are incident with 𝑣, that is, 𝐸_𝐺(𝑣)={𝑒∈𝐸(𝐺)∣𝑣∈𝑉(𝑒)}. Raghunathan et al. [3] characterized the circuits of the graph 𝐺_𝑥,𝑦 in terms of circuits of 𝐺 as follows.

Lemma 2.1 (see [3]). Let 𝐺 be a graph and let {𝑥,𝑦} be a pair of adjacent edges of 𝐺. Then a subset 𝐶 of edges of the graph 𝐺_𝑥,𝑦 is a circuit in 𝐺_𝑥,𝑦 if and only if 𝐶 satisfies one of the following conditions:(i)𝐶is a circuit in 𝐺 containing 𝑥 and 𝑦;(ii)𝐶 is a circuit in 𝐺 containing neither 𝑥 nor 𝑦;(iii)𝐶=𝐶₁∪𝐶₂, where 𝐶₁ and 𝐶₂ are edge disjoint circuits of 𝐺 with 𝑥∈𝐶₁, 𝑦∈𝐶₂, and 𝐶₁∪𝐶₂ does not contain a circuit in 𝐺 satisfying either (𝑖) or (𝑖𝑖) above.

Lemma 2.2. Let 𝐺 be a 2-connected graph and 𝑣 be a vertex of 𝐺 with 𝑑(𝑣)≥3 and {𝑥,𝑦}⊆𝐸_𝐺(𝑣) such that the graph 𝐺_𝑥,𝑦 is not 2-connected. Then 𝐺_𝑥,𝑦 is connected and has exactly two pendant blocks. Further, one pendant block contains {𝑥,𝑦}, and the other pendant block contains 𝐸_𝐺(𝑣)−{𝑥,𝑦}.

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

Figure 3 

Lemma 2.3. Let 𝐺 be a 2-connected graph and let 𝑣 be a vertex of 𝐺 with 𝑑(𝑣)≥4 such that 𝐺_𝑥,𝑦 is not 2-connected for all {𝑥,𝑦}⊆𝐸_𝐺(𝑣). Then, for a given {𝑥,𝑦}⊆𝐸_𝐺(𝑣), the graph 𝐺_𝑥,𝑦 is connected and has one cut vertex and two blocks.

Proof. Let {𝑥,𝑦} be a pair of edges incident with 𝑣. By Lemma 2.2, 𝐺_𝑥,𝑦 is connected and has exactly two pendant blocks, say 𝐵₁ and 𝐵₂. We may assume that 𝐵₁ contains {𝑥,𝑦} and 𝐵₂ contains 𝐸_𝐺(𝑣)−{𝑥,𝑦}. As 𝑑(𝑣)≥4, we can choose two edges 𝑧, 𝑤 from 𝐸_𝐺(𝑣)−{𝑥,𝑦}. Let 𝑃₁ and 𝑃₂ be paths in 𝐺_𝑥,𝑦 from 𝑣 to 𝑣_𝑥,𝑦 with {𝑥,𝑧}⊂𝐸(𝑃₁), {𝑦,𝑤}∩𝐸(𝑃₁)=𝜙, {𝑦,𝑤}⊂𝐸(𝑃₂) and {𝑥,𝑧}∩𝐸(𝑃₂)=𝜙. Each of 𝑃₁ and 𝑃₂ corresponds to a cycle in 𝐺. By Lemma 2.1, these cycles are preserved in the graph 𝐺_𝑦,𝑤. Therefore 𝑃_𝑖 is contained in a block of 𝐺_𝑦,𝑤 for 𝑖=1,2. By Lemma 2.2, 𝐺_𝑦,𝑤 has two pendant blocks one containing edges 𝑦, 𝑤 and the other containing 𝐸_𝐺(𝑣)−{𝑦,𝑤}. Hence 𝑃₁and𝑃₂ belong to different pendant blocks of 𝐺_𝑦,𝑤. Hence 𝑃₁ and 𝑃₂ share at most one vertex of 𝐺_𝑦,𝑤. However, 𝑃₁ and 𝑃₂ share all cut vertices of 𝐺_𝑥,𝑦. This implies that 𝐺_𝑥,𝑦 has exactly one cut vertex. Therefore, by Lemma 2.2, 𝐺_𝑥,𝑦 is connected and has exactly two blocks.

Lemma 2.4. Let 𝐺 and 𝑣 be as stated in Lemma 2.3. Then there exists a vertex 𝑢 in 𝐺 such that 𝑢 is the cut vertex of 𝐺_𝑥,𝑦 for all {𝑥,𝑦}⊆𝐸_𝐺(𝑣).

Proof. Let {𝑥,𝑦}⊆𝐸_𝐺(𝑣). By Lemma 2.3, 𝐺_𝑥,𝑦 is connected and has one cut vertex, say 𝑢. Let {𝑧,𝑤}⊆𝐸_𝐺(𝑣). Then, by Lemma 2.3, 𝐺_𝑧,𝑤 is also connected and has two blocks and one cut vertex. It suffices to prove that 𝑢 is the cut vertex of 𝐺_𝑧,𝑤. If {𝑥,𝑦}={𝑧,𝑤}, then there is nothing to prove. Suppose {𝑥,𝑦}≠{𝑧,𝑤}. We may assume that 𝑥∉{𝑧,𝑤} and 𝑦≠𝑤. By Lemma 2.3, 𝐺_𝑥,𝑤 is connected and has two blocks, say 𝐵₁ and 𝐵₂. By Lemma 2.2, we may assume that 𝐵₁ contains {𝑥,𝑤} and 𝐵₂ contains 𝐸_𝐺(𝑣)−{𝑥,𝑤}. Let 𝑒 be an edge of 𝐺 incident with 𝑣 such that 𝑒∉{𝑥,𝑦,𝑤}. Let 𝑃₁ and 𝑃₂ be paths in 𝐺_𝑥,𝑦 from the vertex 𝑣_𝑥,𝑦 to 𝑣 such that {𝑥,𝑤}⊂𝐸(𝑃₁), {𝑦,𝑒}⊂𝐸(𝑃₂), and {𝑥,𝑤}∩𝐸(𝑃₂)=𝜙, {𝑦,𝑒}∩𝐸(𝑃₁)=𝜙. Then each of 𝑃₁ and 𝑃₂ contains all cut vertices of 𝐺_𝑥,𝑦. Therefore 𝑢 is a common vertex of 𝑃₁ and 𝑃₂. Further, each of 𝑃₁ and 𝑃₂ corresponds to a cycle in 𝐺. By Lemma 2.1, these cycles are preserved in the graph 𝐺_𝑥,𝑤 and hence are contained in blocks of 𝐺_𝑥,𝑤. Therefore 𝐵_𝑖 contains 𝑃_𝑖 for 𝑖=1,2. Thus 𝑢 is a common vertex of 𝐵₁ and 𝐵₂. This implies that 𝑢 is a cut vertex of 𝐺_𝑥,𝑤. By Lemma 2.3, 𝑢 is the only cut vertex of 𝐺_𝑥,𝑤. Let 𝑄₁ and 𝑄₂ be paths in 𝐺_𝑥,𝑤 from 𝑣 to 𝑣_𝑥,𝑤 such that {𝑧,𝑤}⊆𝐸(𝑄₁) and 𝑥∈𝐸(𝑄₂)−𝐸(𝑄₁). Then 𝑄_𝑖 contains the cut vertex 𝑢 and, further, it corresponds to a cycle in 𝐺 for 𝑖=1,2. By Lemma 2.1, 𝑄_𝑖 corresponds to a cycle of the graph 𝐺_𝑧,𝑤 and hence is contained in a block of 𝐺_𝑧,𝑤 for 𝑖=1,2. These cycles are contained in different blocks of 𝐺_𝑧,𝑤. By Lemma 2.2, one block of 𝐺_𝑧,𝑤 contains the edges 𝑧, 𝑤 and the other block contains the remaining edges of 𝐺 that are incident with 𝑣. Hence 𝑄₁ and 𝑄₂ belong to different blocks of 𝐺_𝑧,𝑤. As 𝑢 is a common vertex of 𝑄₁ and 𝑄₂, it is a cut vertex of 𝐺_𝑧,𝑤. Thus 𝑢 is the cut vertex of 𝐺_𝑧,𝑤.

Lemma 2.5. Let 𝐺 be a 2-connected graph and 𝑣 be a vertex of 𝐺 with the set of neighbours {𝑣₁,𝑣₂,…,𝑣_𝑘}, where 𝑘≥4. Suppose 𝐺_𝑥,𝑦 is not 2-connected for all {𝑥,𝑦}⊆𝐸_𝐺(𝑣). Then there exists a vertex 𝑢 in 𝐺 such that 𝑉(𝑃)∩𝑉(𝑄)={𝑢} for any 𝑢𝑣_𝑖-path 𝑃 and 𝑢𝑣_𝑗-path 𝑄 in 𝐺−𝑣 with 𝑖≠𝑗.

Proof. By Lemma 2.4, there exists a vertex 𝑢 in 𝐺 such that it is the cut vertex of 𝐺_𝑥,𝑦 for all {𝑥,𝑦}⊆𝐸_𝐺(𝑣). Let 𝑃 be a 𝑢𝑣_𝑖-path and 𝑄 be a 𝑢𝑣_𝑗-path in 𝐺−𝑣 with 𝑖≠𝑗. We prove that 𝑉(𝑃)∩𝑉(𝑄)={𝑢}. If 𝑃 or 𝑄 is a trivial graph, then there is nothing to prove. Assume that |𝐸(𝑃)|≥1 and |𝐸(𝑄)|≥1. Without loss of generality, we may assume that 𝑣_𝑖=𝑣₁ and 𝑣_𝑗=𝑣₂. Let 𝑒₁=𝑣𝑣₁ and 𝑒₂=𝑣𝑣₂. Then 𝐺_𝑒1,𝑒2 is connected and has two blocks, say 𝐵₁ and 𝐵₂ (see Figure 4). By Lemma 2.2, we may assume that 𝐵₁ contains {𝑒₁,𝑒₂} and 𝐵₂ contains 𝐸_𝐺(𝑣)−{𝑒₁,𝑒₂}. Since 𝑢 is the cut vertex of 𝐺_𝑒1,𝑒2, the paths 𝑃,𝑄 are contained in 𝐵₁. Let 𝑒₃=𝑣𝑣₃ and 𝑒₄=𝑣𝑣₄. Let 𝑃₁ be a 𝑢𝑣-path in 𝐵₂ containing the edge 𝑒₃ but avoiding 𝑒₄. Let 𝑃₂ be an 𝑢𝑣-path in 𝐵₂ containing 𝑒₄ and avoiding 𝑒₃. Then 𝑉(𝑃_𝑘)∩𝑉(𝑃)=𝑉(𝑃_𝑘)∩𝑉(𝑄)={𝑢} for 𝑘=1,2. Let 𝐶₁=𝑃₁∪𝑃∪𝑒₁ and 𝐶₂=𝑃₂∪𝑄∪𝑒₂. Then each of 𝐶₁ and 𝐶₂ corresponds to a cycle in 𝐺. Further, 𝐶₁ contains 𝑒₁, 𝑒₃ and 𝐶₂ contains 𝑒₂, 𝑒₄. Therefore, by Lemma 2.1, 𝐶₁ and 𝐶₂ correspond to cycles in 𝐺_𝑒1,𝑒3. By Lemmas 2.2 and 2.3, 𝐺_𝑒1,𝑒3 has exactly two blocks one of them contains 𝐶₁ and the other contain 𝐶₂. Hence 𝐶₁ and 𝐶₂ can share at most one vertex. This implies that 𝑃 and 𝑄 can share at most one vertex. Thus 𝑉(𝑃)∩𝑉(𝑄)={𝑢}.

Figure 4 

Proof of Theorem 1.3. Let 𝐺 be a 2-connected graph and let 𝑣 be a vertex of 𝐺 with 𝑑(𝑣)≥4. Suppose 𝐺_𝑥,𝑦 is not 2-connected for every pair {𝑥,𝑦} of edges incident with 𝑣. Let {𝑣₁,𝑣₂,…,𝑣_𝑟} be the set of neighbours of 𝑣. Let 𝑥₁ and 𝑦₁ be any two edges of 𝐺 incident with 𝑣. We may assume that 𝑥₁=𝑣𝑣₁ and 𝑦₁=𝑣𝑣₂. By Lemma 2.5, there exists a vertex 𝑢 in 𝐺 such that 𝑉(𝑃_𝑖)∩𝑉(𝑃_𝑗)={𝑢} for any 𝑖,𝑗 with 𝑖≠𝑗, where 𝑃_𝑖 is a 𝑢𝑣_𝑖-path and 𝑃_𝑗 is a 𝑢𝑣_𝑗-path in 𝐺−𝑣 (see Figure 5). It is easy to see that 𝑑(𝑢)≥𝑑(𝑣)≥4. If 𝑢=𝑣_𝑖 for some 𝑖, then 𝑃_𝑖 is the trivial graph containing only the vertex 𝑢. If 𝑣₂=𝑢, then set 𝑥₂=𝑦₁=𝑢𝑣. If 𝑣₃=𝑢, then set 𝑦₂=𝑣𝑢. In other cases, 𝑃₂ and 𝑃₃ are nontrivial graphs and hence we can take 𝑥₂=𝑢𝑢₂∈𝐸(𝑃₂) and 𝑦₂=𝑢𝑢₃∈𝐸(𝑃₃). It is easy to see that (𝐺_𝑥1,𝑦1)_𝑥2,𝑦2 is 2-connected.

Figure 5 

Now, we prove Theorem 1.4. Let 𝑑_𝐻(𝑣) denotes the degree of a vertex 𝑣 in a graph 𝐻.

Proof of Theorem 1.4. Let 𝐺=𝐺₀ be an Eulerian 2-connected graph. Suppose 𝐺 is not a cycle. Then 𝐺 has a vertex 𝑣 of a degree of at least 4. By Theorem 1.3, we get a pair {𝑥₁,𝑦₁} of edges incident with 𝑣 such that either 𝐺_𝑥1,𝑦1 is 2-connected or (𝐺_𝑥1,𝑦1)_𝑥2,𝑦2 is 2-connected for some pair {𝑥₂,𝑦₂} of edges of 𝐺 having a common vertex other than 𝑣. Denote this new 2-connected graph by 𝐺₁. If 𝐺₁=𝐺_𝑥1,𝑦1, then 𝑑_𝐺1(𝑣)=𝑑_𝐺(𝑣)−2 and 𝑑_𝐺1(𝑤)=𝑑_𝐺(𝑤) for any 𝑤∈𝑉(𝐺)−{𝑣}. If 𝐺₁=(𝐺_𝑥1,𝑦1)_𝑥2,𝑦2, then 𝑑_𝐺1(𝑣)=𝑑_𝐺(𝑣)−2, 𝑑_𝐺1(𝑢)=𝑑_𝐺(𝑢)−2, and 𝑑_𝐺1(𝑤)=𝑑_𝐺(𝑤) for any 𝑤∈𝑉(𝐺)−{𝑢,𝑣}, where 𝑢 is the common vertex of 𝑥₂ and 𝑦₂ other than 𝑣. Further, the new vertices of 𝐺₁ that are created in the splitting procedure have degree two. Obviously, 𝐺₁ is Eulerian. If 𝐺₁ is not a cycle, then we obtain a 2-connected Eulerian graph 𝐺₂ from 𝐺₁ by applying splitting operation once (or twice) which results in reducing the degree of a vertex (or two vertices) of 𝐺₁ 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 2-connected one finally arrives at a cycle which corresponds to an Eulerian trail of 𝐺. 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 𝑀=𝑀[𝐴] be a binary matroid and suppose 𝑥,𝑦∈𝐸(𝑀). Let 𝐴_𝑥,𝑦 be the matrix obtained from 𝐴 by adjoining the row that is zero everywhere except for the entries of 1 in the columns labeled by 𝑥 and 𝑦. The splitting matroid 𝑀_𝑥,𝑦 is defined to be the vector matroid of the matrix 𝐴_𝑥,𝑦. The transition from 𝑀 to 𝑀_𝑥,𝑦 is called a splitting operation. The splitting operation for binary matroids is also studied in [3–6].

We need the following three results.

Lemma 3.2 (see [3]). If 𝑀(𝐺) denotes the circuit matroid of a graph 𝐺, then for a pair {𝑥,𝑦} of adjacent edges in a graph 𝐺,𝑀(𝐺_𝑥,𝑦)=(𝑀(𝐺))_𝑥,𝑦.

Lemma 3.3 (see [3]). Let 𝑀 be a binary matroid and 𝑥,𝑦∈𝐸(𝑀). Then 𝑀 is Eulerian if and only if 𝑀_𝑥,𝑦 is Eulerian.

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

We obtain the following characterization of connected Eulerian graphic matroids.

Theorem 3.5. Let 𝑀 be a connected graphic matroid. Then 𝑀 is Eulerian if and only if it can be transformed into a circuit 𝐶 through a sequence 𝑀=𝑀₀<𝑀₁<⋯<𝑀_𝑛=𝐶 of connected graphic matroids such that 𝑀_𝑖+1 is obtained from 𝑀_𝑖 by applying splitting operation once or twice.

Proof. Let 𝑀 be a connected graphic matroid. Then 𝑀 is isomorphic to a cycle matroid 𝑀(𝐺) of some graph 𝐺. In view of Theorem 3.4, we may assume that 𝐺 is 2-connected. Suppose 𝑀 is Eulerian. Then the graph 𝐺 is Eulerian. By Theorem 1.4, there is a sequence of 2-connected graphs 𝐺=𝐺₀<𝐺₁<𝐺₂<⋯<𝐺_𝑛 such that 𝐺_𝑛 is a cycle, and 𝐺_𝑖+1 is obtained from 𝐺_𝑖 by applying splitting operation once or twice for 𝑖=0,1,2,…,𝑛−1. Let 𝑀_𝑖=𝑀(𝐺_𝑖) for 𝑖=0,1,…,𝑛. By Theorem 3.4, each 𝑀_𝑖 is connected. It follows from Lemma 3.2 that if 𝐺_𝑖+1 is obtained from 𝐺_𝑖 by applying splitting operation once or twice then 𝑀_𝑖+1 is obtained from 𝑀_𝑖 by applying splitting operation once or twice, respectively. Further, by Lemma 3.3, 𝑀_𝑖 is Eulerian for 𝑖=1,2,…,𝑛.
Conversely, suppose there exists a sequence of connected graphic matroids 𝑀=𝑀₀<𝑀₁<⋯<𝑀_𝑛=𝐶, where 𝑀_𝑖+1 is obtained from 𝑀_𝑖 by applying splitting operation once or twice for 𝑖=0,1,2,…,𝑛−1. Since 𝑀_𝑛=𝐶 is Eulerian, by Lemma 3.3, 𝑀_𝑛−1 is Eulerian. By repeated applications of Lemma 3.3, we see that 𝑀_𝑖 is Eulerian for each 𝑖. Thus 𝑀=𝑀₀ 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.