Analysis on component connectivity of bubble-sort star graphs and burnt pancake graphs☆
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 , where vertices represent processors and edges represent communication links between processors. For a graph , a subgraph obtained from by removing a set of faulty vertices is denoted by . In particular, is called a vertex-cut of a connected graph if becomes disconnected. If is not a complete graph, the connectivity is defined to be the cardinality of a minimum vertex-cut of . By convention, the connectivity of a complete graph with vertices is defined to be . A graph is -connected provided . Accordingly, the connectivity of a graph 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 and/or the component of . As a generalization of the classical connectivity, a notion concerning the number of components associated with the disconnected graph 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 -component cut of a graph is a set of faulty vertices whose removal from results in a graph with at least components. The -component connectivity of a graph , denoted by , is the cardinality of a minimum -component cut of . By the definition, it is obvious that and for every positive integer .
So far, the exact values of -component connectivity are known only for a few classes of networks and almost all of them are about small ’s. For example, Hsu et al. [22] and Zhao et al. [38] dealt with the -dimensional hypercube for and , respectively. Cheng et al. dealt with the -dimensional hierarchical cubic networks for [6], the -dimensional complete cubic networks for [7], and the generalized exchanged hypercubes with for [8]. Zhao and Yang [37] dealt with the -dimensional folded hypercube for and . Zhao et al. [36] dealt with the -dimensional dual cubes for and . In addition, Chang et al. [3], [4] dealt with the alternating group networks for and Guo et al. [16] dealt with the twisted cubes for . Note that the number of vertices of graphs in the above classes is an exponent related to .
In this paper, we study the -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 -extra connectivity for [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 denote the -dimensional bubble-sort star graph. In this paper, we obtain the following results that for , and and for .
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 denote the -dimensional burnt pancake graph. Kaneko [25] proposed a polynomial-time algorithm to construct vertex-disjoint paths from a source to destinations in . He also showed in [26] that for any faulty subset of vertices , the graph still has a fault-free Hamiltonian cycle (resp. a fault-free Hamiltonian path between any pair of non-faulty vertices) when (resp. ). Chin et al. [9] proved that is super spanning connected if and only if . Iwasaki and Kaneko [24] gave a fault-tolerant routing algorithm for . Compeau [11] determined the girth (i.e., the length of a shortest cycle) of . Lai et al. [27] showed that admits the property of mutually independent Hamiltonian cycles. Hu and Liu [23] determined the (conditional) matching preclusion number of . Moreover, Song et al. investigated the classic diagnosability and the conditional diagnosability of under the comparison model [30] and PMC model [31], respectively. In this paper, we obtain the following results that and for , and for .
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 -component connectivity of for . Section 4 determines the -component connectivity of for . The last section contains our conclusion.
Section snippets
Preliminary
Let be a simple and undirected graph with the vertex set and edge set . The subgraph of induced by a vertex set is denoted by . A maximally connected subgraph of is called a component. A component is trivial if it has only one vertex (i.e., a singleton); otherwise, it is nontrivial. For a vertex , the neighborhood of in , denoted by , is the set of vertices adjacent to in . For a subset , the neighborhood of in is defined as
The component connectivity of
An independent set in a graph is a subset of such that any two vertices of are nonadjacent in .
Lemma 3.1 Let be an independent set of for . Then the following assertions hold. If , then . If , then .
Proof For (1), let . By Lemma 2.1(4), and have at most three common neighbors, i.e., . Since is -regular by Lemma 2.1(5), . For (2), let . We prove the result by
The component connectivity of
First, we consider the lower bounds of the number of neighbors for an independent set of as follows.
Lemma 4.1 Let be an independent set of for . Then the following assertions hold. If , then . If , then . If , then .
Proof For (1), let . By Lemma 2.6(2), contains no 4-cycle. Thus, and have at most one common neighbor, and . For (2), let . By Lemma 2.6(2), contains no 6-cycle.
Concluding remarks
In this paper, we study the -component connectivity of BS graphs and BP graphs for . We have shown that results of BS graphs are for , and and for . Also, we have shown that results of BP graphs are and for , and for . So far the problem of determining and for is still unsolved.
Fàbrega and Fiol [12] introduced another measurement of the reliability and fault tolerance
Acknowledgments
We would like to thank the anonymous referees for a number of helpful comments and suggestions.
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2023, Discrete Applied MathematicsCitation Excerpt :Therefore, the connectivity of networks also has some new extensions and generalizations. For instance, super connectivity [11,27], extra connectivity [12,38], component connectivity [2,15,32], spanning connectivity [3], restricted connectivity [38], structure connectivity [29], and generalized connectivity [18]. In this paper, we study the neighbor connectivity and edge neighbor connectivity on the two kinds of graphs, namely pancake graphs and burnt pancake graphs, which will be defined later in Section 2.
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This work was supported by the China Postdoctoral Science Foundation under the project 2018M631322 (M.-M. Gu), by the National Natural Science Foundation of China under the projects 11971054, 11731002 and the 111 Project of China B16002 (R.-X. Hao), and by the Ministry of Science and Technology of Taiwan under the grants MOST-108-2410-H-606-008 (S.-M. Tang) and MOST-107-2221-E-141-001-MY3 (J.-M. Chang).