Pancake graphs: Structural properties and conditional diagnosability

Abstract

Because of the increasing size of multi-processor systems, processor-fault diagnosis has played critical role in measuring reliability. The diagnosability of numerous well-known multiprocessor systems has been widely investigated. The conditional diagnosability is a new measure of diagnosability by restricting an additional condition under which any fault set cannot contain all the neighbors of any node in a system. This study evaluated the conditional diagnosability for pancake graphs in the PMC model. First, several properties of pancake graphs were derived and, based on these properties, the conditional diagnosability of an n-dimensional pancake graph was shown to be 2 for $n=3$ and $8n-21$ for $n\ge 4$.

Appendix

1.1 Proofs of $t_c(\mathcal{P}_3)=2$ and $t_c(\mathcal{P}_4)=11$

As show in Fig. 17, $\mathcal{P}_3$ is not conditionally 3-diagnosable and $\mathcal{P}_4$ is not conditionally 12-diagnosable. Hence, $t_c(\mathcal{P}_3)\le 2$ and $t_c(\mathcal{P}_4) \le 11$. Because the 3-dimensional pancake graph $\mathcal{P}_3$ is isomorphic to the 3-dimensional star graph $S_3$, according to the result shown in Chang et al. (2005), the diagnosability of $S_3$ is 2. Hence, $t(\mathcal{P}_3)= 2$. Moreover, according to Lemma 8, $t_c(\mathcal{P}_3)\ge 2$, which leads $t_c(\mathcal{P}_3)=2$.

Fig. 17

Two indistinguishable conditional-pairs for $\mathcal{P}_3$ and $\mathcal{P}_4$

For $\mathcal{P}_4$, let $F_1, F_2 \subset V(\mathcal{P}_4)$ be an indistinguishable conditional-pair, and let $T = F_1 \cap F_2$. According to Lemma 10, $\mathcal{P}_4 {\setminus } T$ has a component H with $V(H)\subseteq F_1\varDelta F_2$, which implies that $|F_1{\setminus } F_2|\ge \left\lceil \frac{|V(H)|}{2}\right\rceil$ or $|F_2{\setminus } F_1|\ge \left\lceil \frac{|V(H)|}{2}\right\rceil$; hence, $|F_1|\ge |T|+ \left\lceil \frac{|V(H)|}{2}\right\rceil$ or $|F_2|\ge |T|+ \left\lceil \frac{|V(H)|}{2}\right\rceil$. Therefore, only$|T|+ \left\lceil \frac{|V(H)|}{2}\right\rceil \ge 12$ must be proven.

If $\mathcal{P}_4 {\setminus } T$ is connected, let H be the only component in $\mathcal{P}_4 {\setminus } T$. Therefore,

$$\begin{aligned}&\textstyle |T|+\left\lceil \frac{|V(H)|}{2}\right\rceil \\&\quad = \textstyle |T|+\left\lceil \frac{4!-T}{2}\right\rceil \\&\quad = \textstyle \left\lceil \frac{24+T}{2}\right\rceil \\&\quad \ge 12. \end{aligned}$$

Otherwise, $\mathcal{P}_4 {\setminus } T$ is disconnected. Based on the size of |T|, the following scenarios are considered.

Case 1::

$|T|\le 3$.

According to Lemma 4, $\mathcal{P}_4 {\setminus } T$ has a large component and at most one trivial component. According to Lemma 9, the components in $\mathcal{P}_4 {\setminus } T$ are all non-trivial. Hence, $\mathcal{P}_4 {\setminus } T$ is connected, which leads to a contradiction.

Case 2::

$|T|=4$.

According to Lemma 5, $\mathcal{P}_4 {\setminus } T$ has a large component and at most two small components containing up to 2 nodes in total. Because $(F_1,F_2)$ is an indistinguishable conditional-pair, according to Lemma 10, $\mathcal{P}_4 {\setminus } T$ has a component H such that $|V(H)|\ge 8$. Clearly, H is a large component such that $|V(H)|\ge 4! -|T|-2 \ge 18$ and $|T|+\left\lceil \frac{|V(H)|}{2}\right\rceil \ge 4+\left\lceil \frac{18}{2}\right\rceil \ge 13$.

Case 3::

$5 \le |T|\le 6$.

This case can be verified through an exhaustive search using a computer program. For $|T|=5$, $V(H)\ge 16$. Hence, $|T|+\left\lceil \frac{|V(H)|}{2}\right\rceil \ge 5+\left\lceil \frac{16}{2}\right\rceil \ge 13$. For $|T|=6$, $V(H)\ge 12$. Hence, $|T|+\left\lceil \frac{|V(H)|}{2}\right\rceil \ge 6+\left\lceil \frac{12}{2}\right\rceil \ge 12$.

Case 4::

$|T|=7$.

According to Lemma 10, $\mathcal{P}_4 {\setminus } T$ has a component H such that $V(H)\ge 8$ and H contains a path of length 7 S as a subgraph. According to Lemma 6, $|N_{\mathcal{P}_4}(S)|\ge 8$, and $N_{\mathcal{P}_4}(S)$ belongs to $H \cup T$. Because $|T|=7$, at least one node exists in $N_{\mathcal{P}_4}(S)$ that belongs to H, where $|V(H)|\ge 9$. Therefore, $|T|+\left\lceil \frac{|V(H)|}{2}\right\rceil \ge 7+\left\lceil \frac{9}{2}\right\rceil \ge 12$.

Case 5::

$|T|\ge 8$.

Because $V(H)\ge 8$, $|T|+\left\lceil \frac{|V(H)|}{2}\right\rceil \ge 8+\left\lceil \frac{8}{2}\right\rceil \ge 12$.

According to the above all cases, because $|T|+\left\lceil \frac{|V(H)|}{2}\right\rceil \ge 12 > 11$, $|F_1| >11$ or $|F_2| >11$. According to Lemma 7, $\mathcal{P}_4$ is conditionally 11-diagnosable (i.e., $t_c(\mathcal{P}_4) \ge 11$). Hence, $t_c(\mathcal{P}_4)=11$. $\square$