Citation: Lee, C.-M. Remarks on Parameterized Complexity of Variations of the Maximum-Clique Transversal Problem on Graphs. Symmetry 2022, 14, 676. https:// doi.org/10.3390/sym14040676 Academic Editor: Manuel Lafond Received: 6 March 2022 Accepted: 23 March 2022 Published: 24 March 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). symmetry S S Article Remarks on Parameterized Complexity of Variations of the Maximum-Clique Transversal Problem on Graphs Chuan-Min Lee Department of Computer and Communication Engineering, Ming Chuan University, 5 De Ming Road, Guishan District, Taoyuan City 333, Taiwan; joneslee@mail.mcu.edu.tw; Tel.: +886-3-350-7001 (ext. 3432); Fax: +886-3-359-3876 Abstract: With the rapid growth in the penetration rate of mobile devices and the surge in demand for mobile data services, small cells and mobile backhaul networks have become the critical focus of next-generation mobile network development. Backhaul requirements within current wireless networks are almost asymmetrical, with most traffic flowing from the core to the handset, but 5G networks will require more symmetrical backhaul capability. The deployment of small cells and the placement of transceivers for cellular phones are crucial in trading off the symmetric backhaul capability and cost-effectiveness. The deployment of small cells is related to the placement of transceivers for cellular phones. Chang, Kloks, and Lee transformed the placement problem into the maximum-clique transversal problem on graphs. From the theoretical point of view, our paper considers the parameterized complexity of variations of the maximum-clique transversal problem for split graphs, doubly chordal graphs, planar graphs, and graphs of bounded treewidth. Keywords: parameterized complexity; asymmetric networks; signed maximum-clique transversal function; minus maximum-clique transversal function; symmetrical backhaul capability 1. Introduction Fixed/mobile network convergence enhances the competitive advantage of telecom- munications operators/companies. With the rapid growth in the penetration rate of mobile devices and the surge in demand for mobile data services, telecommunications operators are stepping up their pace to actively improve their wireless network infrastructure to cope with the advent of the mobile broadband networks era. They have to extend broadband networks to any place through the seamless connection between fixed and mobile networks, which helps accelerate the deployment time, reduces maintenance costs, and further en- hances market competitiveness. Under the circumstances, small cells and mobile backhaul networks have become the critical focus of next-generation mobile network development. The so-called mobile backhaul network transmits the mobile signal traffic between the base station and the mobile terminal device to the wireless node, and then aggregates and transmits it to the telecommunications core network. Backhaul requirements within current wireless networks are almost asymmetrical, with most traffic flowing from the core to the handset, but 5G networks will require more symmetrical backhaul capability. Furthermore, we must backhaul massive broadband traffic from small cells to the control center when small cells grow very large. Hence, the deployment of small cells and the placement of transceivers for cellular phones are crucial in trading off the symmetric backhaul capability and cost-effectiveness. The deployment of small cells is related to the placement of transceivers for cellular phones. Chang, Kloks, and Lee transformed the placement problem into the maximum-clique transversal problem on graphs [1]. One of their main objectives is as follows. Modern cellular telecommunications systems divide the entire service area into a set of small regions, which are called cells. Cells are generally thought of as hexagonal grids. One standard method used to place transceivers for cellular Symmetry 2022, 14, 676. https://doi.org/10.3390/sym14040676 https://www.mdpi.com/journal/symmetry