A class of additive multiplicative graph functions

AbstractFor a fixed graph G, the capacity function for G, PG, is defined by PG(H) = limn→∞[γG(Hn)]1/n, where γG(H) is the maximum number of disjoint G's in H. In [2], Hsu proved that PK2 is multiplicative or not. In this paper, we prove that PG is multiplicative and additive for some graphs G which include K2. Some properties of PG are also discussed in this paper

Receiver-Initiated Data Collection in Wake-Up Radio Enabled mIoT Networks: Achieving Collision-Free Transmissions by Hashing and Partitioning

To achieve ultra-low energy consumption and decade-long battery lifetime for Internet of Things (IoT) networks, wake-up radio (WuR) appears as an eminent solution. While keeping devices in deep sleep for most of the time, a WuR enabled IoT device can be woken up for data transmission at any time by a wake-up call (WuC). However, collisions happen among WuCs for transmitter-initiated data reporting and among data packets for receiver-initiated data collection. In this article, we propose three novel hashing-based schemes in order to achieve collision-free data transmissions for receiver-initiated data collection. We consider first a simple scenario where all devices in a region of interest are reachable by a WuC message and propose a scheme which facilitates a scheduled time instant for data uploading of each device through a hash function. In the second scenario where IoT devices are distributed across a large region that cannot be covered by a single WuC, we propose two partitioning algorithms to enable data collection across multiple partitions. Furthermore, we extend the scenario by considering device mobility and propose another scheme which improves the partitioning algorithm to deal with mobility. Both analysis and simulations are performed to demonstrate the effectiveness of the proposed schemes.acceptedVersio

Connected Proper Interval Graphs and the Guard Problem in Spiral Polygons

this paper is to study the hamiltonicity of proper interval graphs and applications of these graphs to the guard problem in spiral polygons. The Hamiltonian path (circuit) problem is, given an undirected graph G = (V ,E), to determine whether G contains a Hamiltonian path (circuit). These two problems are well-known NP -complete problems. In the first part of this paper, we shall derive necessary and sufficient conditions for a proper interval graph to have a Hamiltonian path, have a Hamiltonian circuit, and be Hamiltonian-connected, respectively. See [1,4,5]. Let c(G) denote the number of components of G. The scattering number s(G) of a graph G is max fc(G \Gamma S)\Gamma jSj j S ` V and c(G \Gamma S) 6= 1g. The closed neighborhoo

The existence of hyper-L triple-loop networks

AbstractAguiló et al. (Discrete Math. 167/168 (1997) 3–16) have presented some necessary conditions for the existence of hyper-L triple-loop networks. In this paper, we will give necessary and sufficient conditions for the existence of hyper-L triple-loop networks

An efficient algorithm to find a double-loop network that realizes a given L-shape

AbstractDouble-loop networks have been widely studied as an architecture for local area networks. It is well known that the minimum distance diagram of a double-loop network yields an L-shape. Given a positive integer N, it is desirable to find a double-loop network with its diameter being the minimum among all double-loop networks with N nodes. Since the diameter of a double-loop network can be easily computed from its L-shape, one method is to start with a desirable L-shape and then find a double-loop network to realize it. This is a problem discussed by many authors [F. Aguiló, M.A. Fiol, An efficient algorithm to find optimal double loop networks, Discrete Math. 138 (1995) 15–29, R.C. Chan, C.Y. Chen, Z.X. Hong, A simple algorithm to find the steps of double-loop networks, Discrete Appl. Math. 121 (2002) 61–72, C.Y. Chen, F.K. Hwang, The minimum distance diagram of double-loop networks, IEEE Trans. Comput. 49 (2000) 977–979, P. Esqué, F. Aguiló, M.A. Fiol, Double commutative-step diagraphs with minimum diameters, Discrete Math. 114 (1993) 147–157] and it has been open for a long time whether this problem can be solved in O(logN) time. In this paper, we will provide a simple and efficient O(logN)-time algorithm for solving this problem

A simple algorithm to find the steps of double-loop networks

AbstractDouble-loop networks have been widely studied as architecture for local area networks and it is well-known that the minimum distance diagram of a double-loop network yields an L-shape. Given an N, it is desirable to find a double-loop network DL(N;s1,s2) with its diameter being the minimum among all double-loop networks with N stations. Since the diameter can be easily computed from an L-shape, one method is to start with a desirable L-shape and then asks whether there exist s1 and s2 (also called the steps of the double-loop network) to realize it. In this paper, we propose a simple and efficient algorithm to find s1 and s2, which is based on the Smith normalization method of Aguiló, Esqué and Fiol

Receiver-Initiated Data Collection in Wake-Up Radio Enabled mIoT Networks: Achieving Collision-Free Transmissions by Hashing and Partitioning

To achieve ultra-low energy consumption and decade-long battery lifetime for Internet of Things (IoT) networks, wake-up radio (WuR) appears as an eminent solution. While keeping devices in deep sleep for most of the time, a WuR enabled IoT device can be woken up for data transmission at any time by a wake-up call (WuC). However, collisions happen among WuCs for transmitter-initiated data reporting and among data packets for receiver-initiated data collection. In this article, we propose three novel hashing-based schemes in order to achieve collision-free data transmissions for receiver-initiated data collection. We consider first a simple scenario where all devices in a region of interest are reachable by a WuC message and propose a scheme which facilitates a scheduled time instant for data uploading of each device through a hash function. In the second scenario where IoT devices are distributed across a large region that cannot be covered by a single WuC, we propose two partitioning algorithms to enable data collection across multiple partitions. Furthermore, we extend the scenario by considering device mobility and propose another scheme which improves the partitioning algorithm to deal with mobility. Both analysis and simulations are performed to demonstrate the effectiveness of the proposed schemes