Structural Properties of Generalized Exchanged Hypercubes

Abstract

It has been shown that, when a linear number of vertices are removed from a Generalized Exchanged Hypercube (GEH), a generalized version of the interesting exchanged hypercube, its surviving graph consists of a large connected component and smaller component(s) containing altogether a rather limited number of vertices. In this chapter, we further apply the above connectivity result to derive several fault-tolerance related structural parameters for GEH, including its restricted connectivity, cyclic vertex-connectivity, component connectivity, and its conditional diagnosability in terms of the comparison diagnosis model.

Appendix

In this section, we give a proof of $s=3$ for Part 2 of Theorem 1. We first state a number of preliminary results from [9]. (The proof of Lemma 5 was omitted but it is similar to the one for Lemma 4.)

Lemma 4

[9, Lemma  3.3] $GEH(s, t), 1 \le s \le t,$ is $\delta$-maximally connected.

Lemma 5

[9, Lemma  3.4] $GEH(s, t), 2 \le s \le t,$ is $\delta$-tightly super-connected.

Lemma 6

[9, Lemma  4.1] Let $F \subset V(GEH(s, t)), s \in [2, t],$ $|F|\le ks-\frac{k(k-1)}{2},$ there exists Y,  a connected component of GEH(s, t) $-F,$ such that, for all $i \in [0, 2^s+2^t),$ if $C_i-F_i$ is connected, it is a subgraph of Y.

We note that Lemma 5 does not hold for $s=1$ as GEH(1, t) contains $2^t$ Class-0 clusters, each of which is an edge, and two Class-1 clusters, each isomorphic to a $Q_t.$ (Cf. Fig. 2b). Let (u, v) be one of these edges. When $\{u', v'\} \subseteq F,$ $GEH(1, t)-F$ contains (u, v) and other components containing a total of $2^{t+2}-2 \ge 6$ vertices.

We are now ready to prove $s=3$ for Part 2 of Theorem 1. When $s=3,$ $k \in [1, 3].$ We notice that, when $k=1,$ $|F|\le s,$ $GEH(s, t)-F$ is then connected, by Lemma 4. We thus only need to consider the cases of $k=2$ and $k=3.$

For the case of $k=2,$ thus $|F|\le 5$ by Part 2, we need to show that $GEH(3, t)-F, t \ge 3,$ is either connected or contains a large component together with a singleton. By Lemma 5, when $|F| \le 4,$ $GEH(3, t)-F$ is either connected or it consists of a large component and one singleton. Thus, we only need to consider the case of $|F|=5.$

Let $F_i=F \cap V(C_i), i \in [0, 2^s+2^t).$ If, for some $l, |F_l|=5,$ then all the other clusters contain no faulty vertices, thus they are all connected. Clearly $GEH(s, t)-F_l$ will be connected, as well, since every vertex in $C_l-F_l$ is adjacent, via a cross edge, to a vertex located in a connected cluster. If for some l,  $|F_l|=4,$ and the remaining faulty vertex f falls into another cluster, then all the clusters, other than $C_l,$ are connected. $GEH(s, t)-F$ is then either connected or contains a large component and a singleton $u\ (\in V(C_l) \setminus F_l)$, when $C_l$ is isomorphic to $Q_3,$ u is adjacent to f via a cross edge, and all the three neighbors of u in $C_l$ fall into $F_l.$ We now assume that $|F_l|=3,$ when the other clusters collectively hold two faulty vertices, thus all connected by the maximum connectivity of hypercubes, as $s =3.$ If $C_l-F_l$ is connected, so is $GEH(s, t)-F$ by Lemma 6. Otherwise, if $C_l-F_l$ is disconnected, then $C_l$ is isomorphic to $Q_3,$ and $C_l-F_l$ contains a $K_{1, 3}$ and a singleton f. Since the other clusters jointly hold two faulty vertices, this $K_{1, 3}$ must be part of the large component of $GEH(3, t)-F,$ as at least one of its four vertices is adjacent to a non-faulty vertex in this large component. Then, $GEH(3, t), 3 \le t,$ is either connected or contains a large component and one singleton u when u is adjacent to one of the two faulty vertices, while all its three neighbors in $C_l$ form $F_l.$ The other cases are symmetric to the above.

We now turn to the case of $k=3,$ i.e., $|F| \le 6,$ when we have to show that $GEH(3, t)-F, t \ge 3,$ is either connected or contains a large component and small components altogether with at most two vertices. In light of the previous case, we only need to consider the case of $|F|=6.$

If for some $l, |F_l| \ge 4,$ then other clusters, sharing at most two faulty vertices, must be individually connected in the resulting graph by the assumption of $s=3$ and Lemma 4, and belong to the same component, say Y, in the resulting graph by Lemma 6. By definition, those non-faculty vertices of $C_l$ are part of Y. Hence, $GEH(s, t)-F$ is either connected, or contains a large component and smaller ones with at most two vertices, when the remaining up to two vertices in $V(C_l) \setminus C_l$ are adjacent to the faulty vertices in $F \setminus F_l$ via cross edges, while sharing their faulty neighbors in $F_l.$

We now consider the case when, for all l,  $F_l$ contains at most three of these vertices. Since for all l,  $C_l$ is isomorphic to a cube $Q_m, m \ge s\ (=3),$ when $m \ge 4,$ all such $C_l-F_l$’s are connected by Lemma 4, and so is $GEH(s, t)-F,$ by Lemma 6. We thus only need to consider the case when $C_l$ is isomorphic to $Q_3,$ where $|F_l|=3.$

If for some l,  $|F_l|=3,$ and for $j \not = l, |F_j|<3,$ then $C_j-F_j$, $j\not = l,$ will all be connected by Lemma 6. If $C_l-F_l$ is connected, so is $GEH(s, t)-F$ by Lemma 6. Now assume that $C_l-F_l$ is not connected. Notice that $C_l$ is isomorphic to a $Q_3,$ and its surviving graph contains a singleton u and a $K_{1, 3}.$ Since there are only three faulty vertices located outside $C_l,$ and $K_{1, 3}$ contains four vertices, it must be part of a large connected component. Thus, in this case, $GEH(3, t), t \ge 3,$ is either connected or contains a large component and a singleton u,  when u is adjacent to one of these remaining faulty vertices in $F \setminus F_l,$ and all its three neighbors are contained in $F_l.$

We finally consider the subcase that $|F_l|=|F_{l'}|=3,$ where both $C_l$ and $C_{l'}$ are isomorphic to $Q_3,$ when, for $j \not \in \{l, l'\},$ $F_j$ is empty. If both $C_l-F_l$ and $C_{l'}-F_{l'}$ are connected, then GEH(s, t) is also connected by Lemma 6. We now assume, without loss of generality, $C_l-F_l$ is connected, but $C_{l'}-F_{l'}$ is not, when it contains a singleton $u'$ and a $K_{1, 3}.$ $GEH(s, t)-F,$ in this case, is either connected or contains a large component and a singleton $u'$ when it is adjacent to a vertex in $F_l$ and all its three neighbors in $C_l'$ constitute $F_l'.$ For the remaining case, when neither of them is connected, namely, $C_l-F_l$ (respectively, $C_{l'}-F_{l'}$) contains a singleton u (respectively, $u'$) and a $K_{1, 3}.$ By the same token, $GEH(s, t)-F$ is either connected or it contains a large component and smaller component(s) with at most two faulty vertices u and $u',$ when $u'$ (respectively, u) is adjacent to a vertex in $F_l$ (respectively, $F_{l'}$) and all its three neighbors in $C_l'$ (respectively, $C_l$) fall into $F_l'$ (respectively, $F_l$).