Analysis on component connectivity of bubble-sort star graphs and burnt pancake graphs

Abstract

The $\ell$-component connectivity of a graph $G$, denoted by $c{\kappa }_{\ell }\left(G\right)$, is the minimum number of vertices whose removal from $G$ results in a disconnected graph with at least $\ell$ components or a graph with fewer than $\ell$ vertices. This is a natural generalization of the classical connectivity of graphs defined in terms of the minimum vertex-cut. Since this parameter can be used to evaluate the reliability and fault tolerance of a graph $G$ corresponding to a network, determining the exact values of $c{\kappa }_{\ell }\left(G\right)$ is an important issue on the research topic of networks. However, it has been pointed out in Hsu et al. (2012) that determining $\ell$-component connectivity is still unsolved in most interconnection networks even for small $\ell$’s. Let $B{S}_n$ and $B{P}_n$ denote the $n$-dimensional bubble-sort star graph and the $n$-dimensional burnt pancake graph, respectively. In this paper, for $B{S}_n$, we determine the values: $c{\kappa }_3\left(B{S}_n\right)=4n-9$ for $n⩾3$, and $c{\kappa }_4\left(B{S}_n\right)=6n-16$ and $c{\kappa }_5\left(B{S}_n\right)=8n-24$ for $n⩾4$. Similarly, for $B{P}_n$, we determine the values: $c{\kappa }_3\left(B{P}_n\right)=2n-1$ and $c{\kappa }_4\left(B{P}_n\right)=3n-2$ for $n⩾4$, and $c{\kappa }_5\left(B{P}_n\right)=4n-4$ for $n⩾5$.

Introduction

Graph connectivity plays a key role in the study of modern interconnection networks, and usually, such a network is modeled by a connected graph $G=(V,E)$, where vertices represent processors and edges represent communication links between processors. For a graph $G$, a subgraph obtained from $G$ by removing a set $F$ of faulty vertices is denoted by $G-F$. In particular, $F$ is called a vertex-cut of a connected graph $G$ if $G-F$ becomes disconnected. If $G$ is not a complete graph, the connectivity $\kappa \left(G\right)$ is defined to be the cardinality of a minimum vertex-cut of $G$. By convention, the connectivity of a complete graph with $n$ vertices is defined to be $n-1$. A graph $G$ is $k$-connected provided $\kappa \left(G\right)⩾k$. Accordingly, the connectivity of a graph $G$ can be used to assess the vulnerability of the corresponding network and is an important measurement for the reliability and fault tolerance of networks [35].

Moreover, to further analyze the detailed situation of the disconnected graph caused by a vertex-cut, Harary [19] suggested studying the conditional connectivity with additional restrictions on the vertex-cut $F$ and/or the component of $G-F$. As a generalization of the classical connectivity, a notion concerning the number of components associated with the disconnected graph $G-F$ was first introduced by Chartrand et al. [5] (which is called the generalized connectivity) and Sampathkumar [29] (which is called the general connectivity), independently. Hereafter, we use a more suitable term called the component connectivity proposed by Hsu et al. [22]. An $\ell$-component cut of a graph $G$ is a set of faulty vertices whose removal from $G$ results in a graph with at least $\ell$ components. The $\ell$-component connectivity of a graph $G$, denoted by $c{\kappa }_{\ell }\left(G\right)$, is the cardinality of a minimum $\ell$-component cut of $G$. By the definition, it is obvious that $c{\kappa }_2\left(G\right)=\kappa \left(G\right)$ and $c{\kappa }_{\ell +1}\left(G\right)⩾c{\kappa }_{\ell }\left(G\right)$ for every positive integer $\ell$.

So far, the exact values of $\ell$-component connectivity are known only for a few classes of networks and almost all of them are about small $\ell$’s. For example, Hsu et al. [22] and Zhao et al. [38] dealt with the $n$-dimensional hypercube $Q_n$ for $2⩽\ell ⩽n+1$ and $n+2⩽\ell ⩽2n-4$, respectively. Cheng et al. dealt with the $n$-dimensional hierarchical cubic networks $HCN\left(n\right)$ for $2⩽\ell ⩽n+1$ [6], the $n$-dimensional complete cubic networks $CCN\left(n\right)$ for $2⩽\ell ⩽n+1$ [7], and the generalized exchanged hypercubes $GEH(s,t)$ with $s⩽t$ for $2⩽\ell ⩽s+1$ [8]. Zhao and Yang [37] dealt with the $n$-dimensional folded hypercube $F{Q}_n$ for $1⩽\ell ⩽n-1$ and $n⩾8$. Zhao et al. [36] dealt with the $n$-dimensional dual cubes $D_n$ for $1⩽\ell ⩽n-1$ and $n⩾2$. In addition, Chang et al. [3], [4] dealt with the alternating group networks $A{N}_n$ for $\ell =3,4$ and Guo et al. [16] dealt with the twisted cubes for $\ell =3,4$. Note that the number of vertices of graphs in the above classes is an exponent related to $n$.

In this paper, we study the $\ell$-component connectivity of bubble-sort star graphs (BS graphs for short) and burnt pancake graphs (BP graphs for short), which will be defined later in Section 2. Although BS graphs have emerged for more than twenty years and were proposed by Chou et al. [10], the structural properties of such graphs are extensively and intensively studied only recently. Due to that a BS graph is a graph with the composition of edges for two kinds of graphs called star graphs and bubble-sort graphs, as expected, BS graphs have the advantage in diverse connectivities, such as the fault-tolerant maximally local connectivity [2], the $h$-extra connectivity for $1⩽h⩽3$ [18], [33], the 2-good-neighbor connectivity [34], and the strong connectivity allowing processors and communication links to fail simultaneously [32]. Moreover, different types of diagnosability in measuring the reliability of BS graphs were also discussed recently, such as the conditional diagnosability under the MM model and PMC model [17], the 2-extra diagnosability under the PMC model and MM* model [33], the 2-good-neighbor diagnosability under the PMC model and MM* model [34], and the pessimistic diagnosability [15]. Let $B{S}_n$ denote the $n$-dimensional bubble-sort star graph. In this paper, we obtain the following results that $c{\kappa }_3\left(B{S}_n\right)=4n-9$ for $n⩾3$, and $c{\kappa }_4\left(B{S}_n\right)=6n-16$ and $c{\kappa }_5\left(B{S}_n\right)=8n-24$ for $n⩾4$.

Gates and Papadimitriou [14] introduced the well-known burnt pancake problem in 1979. This problem is also known as sorting by prefixing reversals and relates to the construction of networks of parallel processors. Let $B{P}_n$ denote the $n$-dimensional burnt pancake graph. Kaneko [25] proposed a polynomial-time algorithm to construct $n$ vertex-disjoint paths from a source to $n$ destinations in $B{P}_n$. He also showed in [26] that for any faulty subset of vertices $F$, the graph $B{P}_n-F$ still has a fault-free Hamiltonian cycle (resp. a fault-free Hamiltonian path between any pair of non-faulty vertices) when $\left|F\right|⩽n-2$ (resp. $\left|F\right|⩽n-3$). Chin et al. [9] proved that $B{P}_n$ is super spanning connected if and only if $n\ne 2$. Iwasaki and Kaneko [24] gave a fault-tolerant routing algorithm for $B{P}_n$. Compeau [11] determined the girth (i.e., the length of a shortest cycle) of $B{P}_n$. Lai et al. [27] showed that $B{P}_n$ admits the property of $n$ mutually independent Hamiltonian cycles. Hu and Liu [23] determined the (conditional) matching preclusion number of $B{P}_n$. Moreover, Song et al. investigated the classic diagnosability and the conditional diagnosability of $B{P}_n$ under the comparison model [30] and PMC model [31], respectively. In this paper, we obtain the following results that $c{\kappa }_3\left(B{P}_n\right)=2n-1$ and $c{\kappa }_4\left(B{P}_n\right)=3n-2$ for $n⩾4$, and $c{\kappa }_5\left(B{P}_n\right)=4n-4$ for $n⩾5$.

The remaining part of this paper is organized as follows. Section 2 formally gives the definitions of bubble-sort star graphs and burnt pancake graphs. In addition, we introduce some preliminary results that will be used later. Section 3 determines the $\ell$-component connectivity of $B{S}_n$ for $\ell =3,4,5$. Section 4 determines the $\ell$-component connectivity of $B{P}_n$ for $\ell =3,4,5$. The last section contains our conclusion.