Fault tolerant routing in the star and pancake interconnection networks☆

Let $G=(V,E)$ be a connected graph. The h-restricted edge connectivity ${\lambda }_h(G)$ of G is defined as the minimum size $|F|$ of a set F of edges such that $G-F$ is disconnected and every component of $G-F$ contains at least h vertices. G is said to be ${\lambda }_h$-connected if ${\lambda }_h(G)$ exists. Let ${\xi }^h(G)=min\{|\omega (A)|:\mathrm{G}[\mathrm{A}]\mathit{is\; connected\; and}|A|=h\}$, where $\omega (A)$ is the subset of edges having exactly one end node in A and $G[A]$ is the subgraph induced by the node set A. A ${\lambda }_h$-connected graph G is said to be ${\lambda }_h$-optimal if ${\lambda }_h(G)={\xi }^h(G)$. A ${\lambda }_h$-optimal graph G is said to be m-${\lambda }_h$-optimal if $G-F$ is still ${\lambda }_h$-optimal for any edge subset $F\subseteq E(G)$ with $|F|\le m$. The edge fault tolerance of a ${\lambda }_h$-optimal graph G with respect to the ${\lambda }_h$-optimal property, denoted by ${\rho }_h(G)$, is the maximum integer m such that G is m-${\lambda }_h$-optimal. In this paper, we give the lower and upper bounds of ${\rho }_2$ for ${\lambda }_2$-optimal graphs. As applications, we determine exact values of ${\rho }_2$ for two families of networks.

In this paper, we consider the edge fault tolerance of $n$-dimensional burnt pancake graph $B{P}_n$ such that each vertex is incident with at least two fault free edges. Based on this requirement, we show that $B{P}_n$ contains a fault free Hamilton cycle even it has up to $2n-5$ link faults, where $n\ge 3$.

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings, and the conditional matching preclusion number of a graph is the minimum number of edges whose deletion leaves a resulting graph with no isolated vertices that has neither perfect matchings nor almost perfect matchings. In this paper, we find these two numbers for the burnt pancake graphs and show that every optimal (conditional) matching preclusion set is trivial.

The perfect hierarchical hypercube structure was proposed in the literature as a topology for interconnection networks of massively parallel systems. It has the useful ability that it can connect many nodes while retaining a low degree as well as a small diameter. In this paper, we introduce an algorithm solving the node-to-set disjoint-path routing problem in a perfect hierarchical hypercube. Inside a (2^m + m)-perfect hierarchical hypercube, given one source node and a set of k (≤ m + 1) destination nodes, this algorithm finds k disjoint paths between the source node and all destination nodes of lengths at most m2^m + 2^m + 2^m + 4 in O(km2^m) time complexity.

Vertex connectivity and edge connectivity are two important parameters in interconnection networks. Even though they reflect the fault tolerance correctly, they undervalue the resilience of large networks. By the concept of conditional connectivity and super-connectivity, the concept of restricted vertex connectivity and restricted edge connectivity of graphs was proposed by Esfahanian [A.H. Esfahanian, Generalized measures of fault tolerance with application to N-cube networks, IEEE Transactions on Computers 38 (1989) 1586–1591]. Such measures take the resilience of large networks into consideration. In this paper, we propose three families of interconnection networks and discuss their restricted vertex connectivity and restricted edge connectivity. In particular, the hypercubes, twisted-cubes, crossed-cubes, möbius cubes, star graphs, pancake graphs, recursive circulant graphs, and k-ary n-cubes are special cases of these families.