Local Diagnosability
3.4 Conditional Fault Local Diagnosability
In previous section, we know that Qn does not have the strongly local-diagnosable prop-erty, if there are n − 1 faulty edges, all these faulty edges are incident with a single vertex and this vertex is incident with only one fault-free edge. Therefore, we are led to the following question: How many edges can be removed from Qn such that Qn keeps the strongly local-diagnosable property under the condition that each vertex of the faulty
hypercube Qn is incident with at least two fault-free edges? Firstly, we give an exam-ple to show that a faulty hypercube Qn with 3(n − 2) faulty edges may not have the strongly local-diagnosable property, even if each vertex of the faulty hypercube Qn is incident with at least two fault-free edges. As shown in Figure 3.11(a), we take a cy-cle of length four in Qn, n ≥ 3. Let {v, a, b, c} be the four consecutive vertices on this cycle, and F ⊂ E(Qn) be a set of edges, F = F1S F2S F3, where F1 is the set of all edges incident with a except (v, a) and (b, a), F2 is the set of all edges incident with b except (a, b) and (c, b), and F3 is the set of all edges incident with c except (v, c) and strongly local-diagnosable property, if each vertex of Qn− F is incident with at least two fault-free edges and |F | ≤ 3(n − 2) − 1. We need the following results to construct a Type I structure or a Type II structure at a vertex of a faulty hypercube.
b b
Theorem 14 [52] Let G(V, E) be a bipartite graph with bipartition (X, Y ). Then G has a matching that saturates every vertex in X if and only if
|N(S)| ≥ |S|, for all S ⊆ X.
Theorem 15 [52] Let G(V, E) be a bipartite graph. The maximum size of a matching in G equals the minimum size of a vertex cover of G.
Lemma 8 An n-dimensional hypercube Qn has no cycle of length three and any two vertices have at most two common neighbors.
For our discussion later, we need some definitions. Let Qnbe an n-dimensional hyper-cube and F ⊆ E(Qn) be a set of edges. Removing the edges in F from Qn, for a vertex v two, respectively) vertex under v (see Figure 3.12).
b at least two fault-free edges. Removing all the edges in F from Qn, the local diagnosability of each vertex is still equal to its remaining degree under the PMC model.
Proof.
According to Theorem 9, we can concentrate on the construction of Type I structure or Type II structure at each vertex. Consider a vertex v in Qn− F with degQn−F(v) = k.
As shown in Figure 3.12, let BG(v) = (L1(v)S L2(v), E) be the bipartite graph under v. vertex v. By Theorem 9, the local diagnosability of vertex v is equal to k.
Case 2: |M| = k − 1
We shall show that there is a Type II structure of order k at vertex v. As shown in Figure 3.13, let L1(v) = {x1, x2, ..., xk} and let ML2(v) ⊂ L2(v) be the set of vertices remaining degree of y1 is at least three, as shown in Figure 3.14, there exists a Type II structure T2(v; k − 2, 2) of order k at vertex v. By Theorem 9, the local diagnosability of vertex v is equal to k and the result follows. If the remaining degree of y1 is two, the number of faulty edges incident with y1 is n − 2. Next, we divide the case into two subcases: 2.1), both xk and x1 have remaining degree two and 2.2), one of xk and x1 has remaining degree at least three and the other has at least two.
Subcase 2.1: Both xk and x1 have remaining degree two.
This is an impossible case. Since the number of faulty edges incident with xk and x1
is 2(n − 2), the total number of faulty edges is at least 3(n − 2) which is greater than 3(n − 2) − 1, a contradiction.
Subcase 2.2: One of xk and x1 has remaining degree at least three and the other has at least two.
b
Figure 3.13: Illustration for the case 2 of Theorem 16 and Theorem 18.
b
Without loss of generality, assume xk has remaining degree at least three and x1 has remaining degree at least two. Since degQn−F(xk) ≥ 3, there exist at least two vertices in ML2(v) that are the neighbors of vertex xk. Then, we can find a vertex yi ∈ ML2(v) and yi 6= y1, i ∈ {2, 3, ..., k − 1}, such that (xk, yi) ∈ E(BG(v)). Without loss of generality, let (xk, y2) ∈ E(BG(v)). If the remaining degree of y2 is at least three, there exists a Type II structure T2(v; k − 2, 2) of order k at vertex v. By Theorem 9, the local diagnosability of vertex v is equal to k and the result follows. If the remaining degree of y2 is two, the number of faulty edges incident with y2 is n − 2. We then consider two further cases:
Subcase 2.2.1: Vertex x1 has remaining degree two.
This is an impossible case. Since the number of faulty edges incident with x1 is n − 2, the total number of faulty edges is at least 3(n − 2) which is greater than 3(n − 2) − 1, a
contradiction.
Subcase 2.2.2: Vertex x1 has remaining degree at least three.
Since degQn−F(x1) ≥ 3, there exist at least two vertices in ML2(v) that are the neighbors of vertex x1. By Lemma 8, any two vertices of Qn have at most two common neighbors. We can find a vertex yi ∈ ML2(v), yi 6= y1 and yi 6= y2, i ∈ {3, 4, ..., k − 1}, such that (x1, yi) ∈ E(BG(v)). Without loss of generality, let (x1, y3) ∈ E(BG(v)). If the remaining degree of y3 is at least three, there exists a Type II structure T2(v; k − 2, 2) of order k at vertex v. By Theorem 9, the local diagnosability of vertex v is equal to k and the result follows. If the remaining degree of y3 is two, then the number of faulty edges incident with y3 is n − 2, and the total number of faulty edges is at least 3(n − 2) which is greater than 3(n − 2) − 1, a contradiction.
Case 3: |M| ≤ k − 2
We shall see that this is an impossible case. By Theorem 15, the minimum size of a vertex cover of the bipartite graph BG(v) is no greater than k − 2. We take a vertex cover with the minimum size, and let V CL1(v) ⊂ L1(v), V CL2(v) ⊂ L2(v) and V CL1(v)S V CL2(v) be the vertex cover as shown in Figure 3.15. V CL1(v) and V CL2(v) can cover all the edges of BG(v). Let NV CL1(v) = L1(v) − V CL1(v). We claim that the total number of faulty edges is at least (n − 1)|NV CL1(v)| − 2|V CL2(v)|, and this number is greater than 3(n − 2) which is a contradiction. With this claim, the case is impossible.
Now we prove the claim. First, for each vertex x ∈ NV CL1(v), the edges connecting x except (x, v) must be incident with the vertices in V CL2(v). For each vertex y ∈ V CL2(v), by Lemma 8, at most 2 edges connecting y are incident with the vertices in NV CL1(v).
Then, the total number of faulty edges is at least (n − 1)|NV CL1(v)| − 2|V CL2(v)|.
and |V CL2(v)| ≥ 1, we have the following
In summary, aside from those impossible cases, we showed that Qn− F contains either a Type I structure T1(v; k) or a Type II structure T2(v; k − 2, 2) of order k at vertex v. By Theorem 9, removing all the edges in F from Qn, the local diagnosability of each vertex
is still equal to its remaining degree. 2
b
Figure 3.15: Illustration for the case 3 of Theorem 16 and Theorem 18.
By Theorem 16, we have the following corollary.
Corollary 5 Let Qn be an n-dimensional hypercube with n ≥ 3, and F ⊂ E(Qn) be a set of edges, 0 ≤ |F | ≤ 3(n − 2) − 1. Qn− F has the strong local diagnosability property under the PMC model, provided that each vertex of Qn− F is incident with at least two fault-free edges.
Based on the same requirement, we shall show that Sn keeps the strongly local-diagnosable property no matter how many edges are faulty.
Theorem 17 Let Sn be an n-dimensional star graph with n ≥ 3, and F ⊂ E(Sn) be a set of edges. Assume that each vertex of Sn− F is incident with at least two fault-free edges. Removing all the edges in F from Sn, the local diagnosability of each vertex is still equal to its remaining degree under the PMC model.
Proof.
According to Theorem 9, we can concentrate on the construction of the Type I struc-ture T1 at each vertex. Consider a vertex v in Sn − F with degSn−F(v) = k. Let NSn−F(v) = {x1, x2, ..., xk} be the neighborhood of v. Let L2(v) = {y ∈ V (Sn) | there exists a vertex x ∈ NSn−F(v) such that (x, y) ∈ E(Sn)} − {v}. Since each vertex of Sn− F is incident with at least two fault-free edges and Sn has no cycle of length less than six, the maximum size of a matching from NSn−F(v) to L2(v) is equal to k. As a result, there must exist a Type I structure T1(v; k) of order k at vertex v. By Theorem 9, removing all the edges in F from Sn, the local diagnosability of each vertex is still equal to its
remaining degree. 2
By Theorem 17, the following corollary holds.
Corollary 6 Let Sn be an n-dimensional star graph with n ≥ 3, and F ⊂ E(Sn) be a set of edges. Sn keeps the strongly local- diagnosable property under the PMC model no matter how many edges are faulty, provided that each vertex of Sn− F is incident with at least two fault-free edges.
In the end of this section, we consider another condition: each vertex of a faulty hypercube Qn is incident with at least three fault-free edges. Based on this condition, we prove that Qn keeps the strong local diagnosability property no matter how many edges are faulty.
Theorem 18 Let Qn be an n-dimensional hypercube with n ≥ 3, and F ⊂ E(Qn) be a set of edges. Assume that each vertex of Qn− F is incident with at least three fault-free edges. Removing all the edges in F from Qn, the local diagnosability of each vertex is still equal to its remaining degree under the PMC model.
Proof.
According to Theorem 9, we can concentrate on the construction of Type I structure or Type II structure at each vertex. Consider a vertex v in Qn− F with degQn−F(v) = k.
Let BG(v) = (L1(v)S L2(v), E) be the bipartite graph under v. Then, |L1(v)| = k. Let M ⊂ E(BG(v)) be a maximum matching from L1(v) to L2(v). In the following proof, we consider three cases by the size of M: 1) |M| = k, 2) |M| = k − 1 and 3) |M| ≤ k − 2.
Case 1: |M| = k
Since |M| = k and |L1(v)| = k, there exists a Type I structure T1(v; k) of order k at vertex v. By Theorem 9, the local diagnosability of vertex v is equal to k.
Case 2: |M| = k − 1
We will show that there is a Type II structure of order k at vertex v. As shown in Figure 3.13, let L1(v) = {x1, x2, ..., xk} and let ML2(v) ⊂ L2(v) be the set of vertices matched under M, ML2(v) = {y ∈ L2(v) | there exists a vertex x ∈ L1(v) such that (x, y) ∈ M}. So |ML2(v)| = k − 1. Let ML2(v) = {y1, y2, ..., yk−1} and assume vertex xi is matched with vertex yi for each i, 1 ≤ i ≤ k − 1. Then there exists a vertex xk ∈ L1(v), xk is unmatched by M. Since each vertex of Qn− F is incident with at least three fault-free edges, there exists a vertex yi ∈ ML2(v), i ∈ {1, 2, ..., k − 1}, such that (xk, yi) ∈ E(BG(v)). Without loss of generality, let (xk, y1) ∈ E(BG(v)). Since the remaining degree of y1 is at least three, as shown in Figure 3.14, there exists a Type II structure T2(v; k − 2, 2) of order k at vertex v. By Theorem 9, the local diagnosability of vertex v is equal to k and the result follows.
Case 3: |M| ≤ k − 2
We will see that this is an impossible case. By Theorem 15, the minimum size of a vertex cover of the bipartite graph BG(v) is no greater than k − 2. However, we claim that any k − 2 vertices of BG(v) can not cover all the edges of BG(v). With this claim, the case is impossible.
Now we prove this claim. Suppose not, we take a vertex cover with the minimum size, and let V CL1(v) ⊂ L1(v), V CL2(v) ⊂ L2(v) and V CL1(v)S V CL2(v) be the vertex cover as shown in Figure 3.15. V CL1(v) and V CL2(v) can cover all the edges of BG(v). Since
|V CL1(v)| + |V CL2(v)| ≤ k − 2, we rewrite this inequality into the following equivalent form: 2(k − |V CL1(v)|) ≥ 2(|V CL2(v)| + 2). Let NV CL1(v) = L1(v) − V CL1(v). Since each vertex of Qn − F is incident with at least three fault-free edges, for each vertex x ∈ NV CL1(v), aside from the edge (x, v), at least 2 edges connecting x must be incident with the vertices in V CL2(v). So the total number of edges incident with the vertices in V CL2(v) is at least 2|NV CL1(v)|. For each vertex y ∈ V CL2(v), by Lemma 8, at most 2 edges connecting y are incident with the vertices in NV CL1(v). So the total number of edges incident with the vertices in NV CL1(v) is at most 2|V CL2(v)|. Compare the lower bound 2|NV CL1(v)| and the upper bound 2|V CL2(v)|. We have the following inequality
2|NV CL1(v)| = 2(k − |V CL1(v)|)
≥ 2(|V CL2(v)| + 2) > 2|V CL2(v)|.
The lower bound 2|NV CL1(v)| is greater than the upper bound 2|V CL2(v)|. It means that some edges are not covered by V CL1(v) or V CL2(v) in BG(v). Thus, our claim follows.
In Case 1, Qn− F contains a Type I structure T1(v; k) of order k at vertex v. In Case 2, Qn− F contains a Type II structure T2(v; k − 2, 2) of order k at vertex v. We also proved that Case 3 is impossible. By Theorem 9, removing all the edges in F from Qn, the local diagnosability of each vertex is still equal to its remaining degree. 2
By Theorem 18, the following corollary holds.
Corollary 7 Let Qn be an n-dimensional hypercube with n ≥ 3, and F ⊂ E(Qn) be a set of edges. Qn keeps the strong local diagnosability property under the PMC model no
matter how many edges are faulty, provided that each vertex of Qn− F is incident with at least three fault-free edges.