A Counterexample for the Proof of Implication Conjecture ... - CiteSeerX
A Counterexample for the Proof of Implication Conjecture on Independent Spanning Trees Abishek Gopalan, Srinivasan Ramasubramanian Department of Electrical and Computer Engineering University of Arizona {abishek, srini}@ece.arizona.edu
Abstract — Khuller and Schieber (1992) in [1] developed a constructive algorithm to prove that the existence of k-vertex independent trees in a k-vertex connected graph implies the existence of k-edge independent trees in a k-edge connected graph. In this paper, we show a counterexample where their algorithm fails. 1. Introduction A set of k spanning trees rooted at vertex r are said to be vertex(edge)-independent if the paths from any vertex v(6= r) to r on the k trees are mutually vertex(edge)-disjoint. In [2], Zehavi and Itai posed two conjectures. Vertex Conjecture: Any k-vertex connected graph has k-vertex independent spanning trees rooted at an arbitrary vertex r. Edge Conjecture: Any k-edge connected graph has k-edge independent spanning trees rooted at an arbitrary vertex r. The authors also posed the following question, which we refer to as the Implication Conjecture. It would be interesting to show that either the vertex conjecture implies the edge conjecture, or vice versa. In [1], Khuller and Schieber developed a constructive algorithm to prove that the existence of k-vertex independent trees in a k-vertex connected graph implies the existence of k-edge independent trees in a k-edge connected graph. In this paper, we show a counterexample where their algorithm fails. 2. Counterexample Consider a k-edge connected graph G(V, E). The technique developed in [1] constructs k-edge independent spanning trees in three steps. 1. Transform the given k-edge connected graph G into a k-vertex connected graph, denoted by G0 (V 0 , E 0 ). 2. Assume that k-vertex independent trees are given in G0 . 3. Compute the edge independent trees in G from the vertex independent trees in G0 . Consider the example network shown in Fig. 1. The graph is 3-edge connected. Let R be the root node. Step 1. For each vertex v in G, there are k vertices v 1 through v k in G0 . They are referred to as node vertices and denoted by group(v). For each edge e in G, there is a vertex `(e) in G0 . They are referred to as edge vertices. The edges in G0 are defined as follows. For each edge e connected to a vertex v in G, there are edges from every v j ∈ group(v) to `(e) in G0 . If G is k-edge connected, then G0 is k-vertex connected. Fig. 2
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shows the transformed graph for our example. Note that node vertices of R are shown twice for drawing convenience. Step 2. Assume a set of k-vertex independent spanning trees on the transformed graph. For our example, Figs. 3, 4, and 5 show the three vertex independent trees, T10 , T20 , and T30 , respectively. The trees are rooted at one of the expanded vertices of R, namely R1 . Step 3. Let Pj0 [v 1 , r1 ] denote the path from vertex v 1 to root r1 on tree Tj0 , where j = 1, 2, . . . , k. Let T1 through Tk denote the k-edge independent trees to be constructed in G. According to [1], the parent of vertex v in tree Tj is defined as follows: Let v f (j) be the last vertex on the path Pj0 [v 1 , r1 ] that belongs to group(v). (Clearly such a vertex exists since v 1 is in group(v) and r1 is not.) Let the outgoing edge from v f (j) on Pj0 be (v f (j) , `(em )), for em = (v, u) in G. Then, the parent of v in Tj is defined to be u. Following the above procedure for our example, we obtain three trees T1 through T3 as shown in Figs. 6, 7, and 8, respectively. Observe that the path from W to R on T1 uses edge U –V in the direction V →U , while the path on T2 uses the same edge in the direction U →V . The same holds true for the tree paths of node X to the root as well. Thus, as the tree paths from a node to the root of the spanning trees are not edge disjoint, the trees T1 and T2 are not edge independent. 3. Analysis Let P1 through Pk denote the paths from vertex v to root r on trees T1 through Tk . Lemma 2.3 in [1] claims that the paths P1 through Pk are mutually edge-disjoint. The proof states: Assume that there are two paths P1 [v, r] and P2 [v, r] that use the same edge e. This implies that both paths P10 [v 1 , r1 ] and P20 [v 1 , r1 ] use the same vertex `(e), contradicting the assumption that the paths P10 [v 1 , r1 ] and P20 [v 1 , r1 ] are internally vertex disjoint. The highlighted statement shows the flaw in the proof. In our example, P1 [W, R] and P2 [W, R] share the edge U –V . However, P10 [W 1 , R1 ] and P20 [W 1 , R1 ] are internally vertex disjoint. 4. Conclusion We provided a counterexample that illustrates the failing of the previously established result in [1]. As a consequence, the implication conjecture posed by Zehavi and Itai still remains an open problem. Acknowledgment The research developed in this paper is funded in part by the National Science Foundation under the grant 1117274. References [1] S. Khuller, B. Schieber, On independent spanning trees, Information Processing Letters 42 (6) (1992) 321 – 323. doi:10.1016/0020-0190(92)90230-S. [2] A. Zehavi, A. Itai, Three tree-paths, doi:10.1002/jgt.3190130205.
Journal of Graph Theory 13 (2) (1989) 175–188.
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Figure 2: Transformed graph.
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Figure 3: Vertex independent spanning tree T10 . R1
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Figure 4: Vertex independent spanning tree T20 .
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Figure 5: Vertex independent spanning tree T30 .
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Abstract — Khuller and Schieber (1992) in [1] developed a constructive algorithm to prove that the existence of k-vertex independent trees in a k-vertex connected graph implies the existence of k-edge independent trees in a k-edge connected graph. In this paper, we show a counterexample where their algorithm fails. 1. Introduction A set of k spanning trees rooted at vertex r are said to be vertex(edge)-independent if the paths from any vertex v(6= r) to r on the k trees are mutually vertex(edge)-disjoint. In [2], Zehavi and Itai posed two conjectures. Vertex Conjecture: Any k-vertex connected graph has k-vertex independent spanning trees rooted at an arbitrary vertex r. Edge Conjecture: Any k-edge connected graph has k-edge independent spanning trees rooted at an arbitrary vertex r. The authors also posed the following question, which we refer to as the Implication Conjecture. It would be interesting to show that either the vertex conjecture implies the edge conjecture, or vice versa. In [1], Khuller and Schieber developed a constructive algorithm to prove that the existence of k-vertex independent trees in a k-vertex connected graph implies the existence of k-edge independent trees in a k-edge connected graph. In this paper, we show a counterexample where their algorithm fails. 2. Counterexample Consider a k-edge connected graph G(V, E). The technique developed in [1] constructs k-edge independent spanning trees in three steps. 1. Transform the given k-edge connected graph G into a k-vertex connected graph, denoted by G0 (V 0 , E 0 ). 2. Assume that k-vertex independent trees are given in G0 . 3. Compute the edge independent trees in G from the vertex independent trees in G0 . Consider the example network shown in Fig. 1. The graph is 3-edge connected. Let R be the root node. Step 1. For each vertex v in G, there are k vertices v 1 through v k in G0 . They are referred to as node vertices and denoted by group(v). For each edge e in G, there is a vertex `(e) in G0 . They are referred to as edge vertices. The edges in G0 are defined as follows. For each edge e connected to a vertex v in G, there are edges from every v j ∈ group(v) to `(e) in G0 . If G is k-edge connected, then G0 is k-vertex connected. Fig. 2
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shows the transformed graph for our example. Note that node vertices of R are shown twice for drawing convenience. Step 2. Assume a set of k-vertex independent spanning trees on the transformed graph. For our example, Figs. 3, 4, and 5 show the three vertex independent trees, T10 , T20 , and T30 , respectively. The trees are rooted at one of the expanded vertices of R, namely R1 . Step 3. Let Pj0 [v 1 , r1 ] denote the path from vertex v 1 to root r1 on tree Tj0 , where j = 1, 2, . . . , k. Let T1 through Tk denote the k-edge independent trees to be constructed in G. According to [1], the parent of vertex v in tree Tj is defined as follows: Let v f (j) be the last vertex on the path Pj0 [v 1 , r1 ] that belongs to group(v). (Clearly such a vertex exists since v 1 is in group(v) and r1 is not.) Let the outgoing edge from v f (j) on Pj0 be (v f (j) , `(em )), for em = (v, u) in G. Then, the parent of v in Tj is defined to be u. Following the above procedure for our example, we obtain three trees T1 through T3 as shown in Figs. 6, 7, and 8, respectively. Observe that the path from W to R on T1 uses edge U –V in the direction V →U , while the path on T2 uses the same edge in the direction U →V . The same holds true for the tree paths of node X to the root as well. Thus, as the tree paths from a node to the root of the spanning trees are not edge disjoint, the trees T1 and T2 are not edge independent. 3. Analysis Let P1 through Pk denote the paths from vertex v to root r on trees T1 through Tk . Lemma 2.3 in [1] claims that the paths P1 through Pk are mutually edge-disjoint. The proof states: Assume that there are two paths P1 [v, r] and P2 [v, r] that use the same edge e. This implies that both paths P10 [v 1 , r1 ] and P20 [v 1 , r1 ] use the same vertex `(e), contradicting the assumption that the paths P10 [v 1 , r1 ] and P20 [v 1 , r1 ] are internally vertex disjoint. The highlighted statement shows the flaw in the proof. In our example, P1 [W, R] and P2 [W, R] share the edge U –V . However, P10 [W 1 , R1 ] and P20 [W 1 , R1 ] are internally vertex disjoint. 4. Conclusion We provided a counterexample that illustrates the failing of the previously established result in [1]. As a consequence, the implication conjecture posed by Zehavi and Itai still remains an open problem. Acknowledgment The research developed in this paper is funded in part by the National Science Foundation under the grant 1117274. References [1] S. Khuller, B. Schieber, On independent spanning trees, Information Processing Letters 42 (6) (1992) 321 – 323. doi:10.1016/0020-0190(92)90230-S. [2] A. Zehavi, A. Itai, Three tree-paths, doi:10.1002/jgt.3190130205.
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Figure 3: Vertex independent spanning tree T10 . R1
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Figure 4: Vertex independent spanning tree T20 .
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Figure 5: Vertex independent spanning tree T30 .
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