Strongly Local-diagnosable Property

In this section, we use hypercube as an example to introduce our concept of the strongly local-diagnosable property. In previous section, we presented two sufficient conditions, Theorem 7 and Theorem 8, for identifying the local diagnosability of a vertex. It seems that identifying the local diagnosability of a vertex is the same as counting its degree.

We give an example to show that this is not true in general. As shown in Figure 3.7, we take a vertex v in two-dimensional hypercube Q2, let F1 = {v, 1} and F2 = {2, 3} with

|F₁| = 2 and |F₂| = 2. It is a simple matter to check that (F₁, F2) is an indistinguishable pair. Hence tl(v) 6= deg(v) = 2. We then propose the following two concepts.

bb bb

F

F

v

1 2

3

Figure 3.7: An indistinguishable pair (F₁, F₂) in Q₂.

Definition 7 Let G(V, E) be a graph and v ∈ V be a vertex. Vertex v has the strongly local-diagnosable property if the local diagnosability of vertex v is equal to its degree.

Definition 8 Let G(V, E) be a graph. G has the strongly local-diagnosable property if, every vertex in the graph G has the strongly local- diagnosable property.

Following Definition 7, Definition 8, Theorem 10 and Theorem 11 imply the following two propositions.

Proposition 4 Let Qn be an n-dimensional hypercube, n ≥ 3. Qn has the strongly local-diagnosable property under the PMC model.

Proposition 5 Let Sn be an n-dimensional star graph, n ≥ 3. Sn has the strongly local-diagnosable property under the PMC model.

We now consider a system which is not vertex-symmetric. Let G(V, E) be a graph and F ⊂ E(G) be a set of edges. Removing the edges in F from G, the degree of each vertex in the resulting graph G − F is called the remaining degree of v, and is denoted by deg_G−F(v). We consider a faulty hypercube Qn with a faulty set F ⊂ E(Qn), n ≥ 3. We shall prove that Qnhas the strongly local-diagnosable property even if it has up to (n − 2) faulty edges. The number n − 2 is optimal in the sense that a faulty hypercube Qn cannot be guaranteed to have this strong property if there are n − 1 faulty edges. As shown in Figure 3.8, we take a vertex v ∈ V (Qn) and a vertex x which is an adjacent neighbor of v. of each vertex is still equal to its remaining degree under the PMC model.

Proof.

We use Theorem 9 to prove this result, and we shall construct a Type I structure at each vertex. We prove this by induction on n. For n = 3, 0 ≤ |F | ≤ 1, if |F | = 0,

it is clear that Q₃ contains a Type I structure T₁(v; 3) of order 3 at every vertex. If

|F | = 1, a three-dimensional hypercube Q3 with one missing edge is shown in Figure 3.9.

It is a routine work to see that every vertex has a Type I structure T1(v; k) of order k at it, where k is the remaining degree of the vertex. As the inductive hypothesis, we assume that the result is true for Qn−1, 0 ≤ |F | ≤ (n − 1) − 2, for some n ≥ 4. Now we consider Qn, 0 ≤ |F | ≤ n − 2. If |F | = 0, refer to the proof of Theorem 10, Qn

contains a Type I structure T1(v; n) of order n at every vertex. If 1 ≤ |F | ≤ n − 2, we choose an edge in F , the edge is in some dimension, decomposing Qn into two subcubes Q⁰_n−1 and Q¹_n−1 by this dimension, such that the edge is a crossing edge. Consider a vertex v ∈ V (Qn). Let F0 = FT E(Q⁰_n−1), 0 ≤ |F0| ≤ (n − 3) and F1 = F T E(Q¹_n−1), diagnosability of each vertex is still equal to its remaining degree. 2

bc bc

Figure 3.9: Q3 with one missing edge. The number labeled on each vertex represents its local diagnosability.

We have the following corollary.

Corollary 3 Let Qn be an n-dimensional hypercube with n ≥ 3, and F ⊂ E(Qn) be a set of edges, 0 ≤ |F | ≤ n − 2. Then, Qn− F has the strongly local-diagnosable property under the PMC model.

We now consider a faulty star graph Snwith a faulty set F ⊂ E(Sn), n ≥ 3. Similarly, we shall prove that Sn has the strongly local-diagnosable property even if it has up to (n − 3) faulty edges and the number (n − 3) is also optimal.

Theorem 13 Let Sn be an n-dimensional star graph with n ≥ 3, and F ⊂ E(Sn) be a set of edges, 0 ≤ |F | ≤ n − 3. 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.

We prove this result by constructing a Type I structure T₁ at each vertex. We prove this by induction on n. For n = 3, |F | = 0, it is clear that S3 contains a Type I structure T1(v; 2) of order 2 at every vertex. As the inductive hypothesis, we assume that the result is true for Sn−1, 0 ≤ |F | ≤ (n − 1) − 3, for some n ≥ 4. Now we consider Sn, 0 ≤ |F | ≤ n − 3. If |F | = 0, refer to the proof of Theorem 11, Sn contains a Type I structure T₁(v; n − 1) of order n − 1 at every vertex. If 1 ≤ |F | ≤ n − 3, we choose an edge e ∈ F in some dimension. The star graph can be decomposed into n subgraphs Sn^{1}, Sn^{2}, ..., and Sn^{n}. By the symmetric property of Sn, we may assume that e is a in F from Sn, the local diagnosability of each vertex is still equal to its remaining degree.

2

With Theorem 13, we have the following corollary.

Corollary 4 Let Sn be an n-dimensional star graph with n ≥ 3, and F ⊂ E(Sn) be a set of edges, 0 ≤ |F | ≤ n − 3. Then, Sn− F has the strongly local-diagnosable property under

the PMC model.

We now give an example to show that an n-regular graph G(V, E) has the strong local diagnosability property, but it may not keep this strong property after removing n − 2 edges from G. For example, a 3-regular graph is shown in Figure 3.10(a). The degree of each vertex is 3 and there exists a Type I structure T1(v; 3) of order 3 at each vertex. By Theorem 9, Definition 7 and Definition 8, this graph has the strong local diagnosability property. Let F = {(2, 3)} be a set of one single edge, G − F is shown in Figure 3.10 (b). The vertex u does not have the strong local diagnosability property. The reason is as follows. Let F1 = {u, 1, 4} and F2 = {1, 2, 4} with |F1| ≤ 3, |F₂| ≤ 3. Since there is no edge between V (G) − (F₁S F2) and F₁∆F₂, by Lemma 4, (F₁, F₂) is an indistinguishable pair. Therefore, the local diagnosability of vertex u is at most 2 which is smaller than its degree.

b bb bb bb b b bb bb bb b

(a) (b)

1 2 3

u

1 2 3

u

Figure 3.10: A 3-regular graph without the strong local diagnosability property after removing one edge.