A decomposition of complete bipartite graphs into edge-disjoint subgraphs with star components

Egawa, Yoshimi; Fukuda, Tsukasa; Nagoya, Seiichiro; Urabe, Masatsugu

doi:10.1016/0012-365x(86)90190-1

Cited by 12 publications

(11 citation statements)

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“…The following construction, which uses [M2 + 2] def wavelengths without wavelength changing is adapted 2 d= {(n, mn), (x, y) I n x, m y} (4) from [3]. The construction supports p-permutation We will show that at least [M+]l wavelengths are routing for all 0 < p < 1.…”

Section: Theorem 1 Let Fs(m P) Be the Minimum Number Of Waveotherwismentioning

confidence: 99%

“…vi, and v,,t ple network without wavelength changing to a previously are known as the transmitting and receiving tuning states, solved graph coloring problem [3]. The added complexi-respectively.…”

Section: Theorem 1 Let Fs(m P) Be the Minimum Number Of Waveotherwismentioning

confidence: 99%

“…4 is functionalceivers and that a network which supports permutation ly equivalent to an 1 x F switch, the state of the switch routing is called a connector. Let F(M, S) be the minimum number of wavelengths to do permutation routing 3 This is the construction discussed in section III. being determined by the wavelength of the transmitter.…”

Section: This Is a Contradiction F'(t S) > (1 + E)mentioning

confidence: 99%

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On the number of wavelengths and switches in all-optical networks

Barry

Humblet

1994

IEEE Trans. Commun.

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Abstract-We consider optical networks using wave-a signal launched from a transmitter may arrive at a valength division multiplexing, where the path a signal riety of receivers on many different wavelengths and/or takes is determined by the network switches, the wavearrive at a receiver on several different wavelengths. length of the signal, and the location the signal originated. Therefore, a signal is routed through a combination of circuit switching and wavelength routing (asthe number of available wavelengths, 1 it will be necessary signing it a wavelength). We present a bound on the to simultaneously assign many transmitters the same waveminimum number of wavelengths needed based on the length. Since two signals using the same wavelength canconnectivity requirements of the users and the number not travel over the same fiber simultaneously, collisions of switching states. In addition, we present a lower bound on the number of switching states in a network using a combination of circuit switching, wavelength must insure that signals do not collide at any intended rerouting, and frequency changing. The bounds hold for ceiver. That is, if receiver m is listening to wavelength A all networks with switches, wavelength routing, and at time t, we must insure that only one signal assigned to wavelength changing devices.Several examples are A at time t arrives at receiver m. If two or more arrive, we presented including a network with near optimal wavelength re-use. say there is contention. In this paper we present bounds, some of which are shown to be tight, on the number of wavelengths required under various user connectivity re-I. INTRODUCTION quirements. The lower bounds allow the possibility of re-We consider all-optical networks (AONs) using wavearranging the wavelengths assigned to active sessions to length division multiplexing, circuit switching and waveaccommodate new session requests. None of the construclength routing (A-routing for short). An advantage of A-tions require rearranging. routing is spatial re-use of wavelengths. We show thatIn section II, we formalize the network model and user there is a limit to the possible amount of wavelength reconnectivity requirements.In section III, we analyze two use.special types of passive A-routing networks. We show that In a A-routing network, the path a signal takes is a funcfor these special cases, the number of wavelengths must be tion of the the wavelength of the signal and the location on the order of the number of active sessions for a broad of the signal transmitter. If the signal paths are under class of user connectivity requirements. control of the network, e.g. through the use of switches We then relax the wavelength routing restrictions used or dynamic wavelength routing devices, we say that the in section III and consider A-routing networks with arnetwork is configurable. Otherwise we say the network is bitrary topology, wavelength changers, and switches. A passive or fixed. In a configurable network, the wavelength lower bound on the number of wave...

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Section: Theorem 1 Let Fs(m P) Be the Minimum Number Of Waveotherwismentioning

confidence: 99%

Section: Theorem 1 Let Fs(m P) Be the Minimum Number Of Waveotherwismentioning

confidence: 99%

Section: This Is a Contradiction F'(t S) > (1 + E)mentioning

confidence: 99%

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On the number of wavelengths and switches in all-optical networks

Barry

Humblet

1994

IEEE Trans. Commun.

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“…The star arboricity of a graph G is the minimum number of star forests whose union contains all edges of G. Bounds on star arboricity have been established for several classes of graphs including planar graphs [3,4,11,14].…”

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confidence: 99%

Approximating the Spanning Star Forest Problem and Its Application to Genomic Sequence Alignment

Nguyen¹,

Shen²,

Hou³

et al. 2008

SIAM J. Comput.

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Abstract. This paper studies the algorithmic issues of the spanning star forest problem. We prove the following results: (1) There is a polynomial-time approximation scheme for planar graphs; (2) there is a polynomial-time -approximation algorithm for graphs; (3) it is NP-hard to approximate the problem within ratio 259 260 + for graphs; (4) there is a linear-time algorithm to compute the maximum star forest of a weighted tree; (5) there is a polynomial-time -approximation algorithm for weighted graphs. We also show how to apply this spanning star forest model to aligning multiple genomic sequences over a tandem duplication region.

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“…The independence numbers were determined for the following graphs: the Cartesian product of two odd cycles [8], the direct product of two paths, or two cycles, or a path and a cycle [10], and some specific family of circulant graphs [13]. The star arboricities were studied for the following graphs: complete bipartite graphs [6], [7], [15], complete regular multipartite graphs [3], cubes [15], crowns [11], and planar graphs [2], [9].…”

Section: Introductionmentioning

confidence: 99%