ISBN 978-83-942947-7-9 PORTAL CZM 2015 INFLUENCE OF A VERTEX REMOVING ON THE CONNECTED DOMINATION NUMBER – APPLICATION TO AD-HOC WIRELESS NETWORKS ADRIANA DAPENA, MAGDA DETTLAFF, MAGDALENA LEMA ´ NSKA AND MARIA JOSE SOUTO-SALORIO ABSTRACT. A minimum connected dominating set (MCDS) can be used as virtual backbone in ad-hoc wireless networks for efficient routing and broadcasting tasks. To find the MCDS is an NP- complete problem even in unit disk graphs. Many suboptimal algorithms are reported in the literature to find the MCDS using local information instead to use global network knowledge, achieving an important reduction in complexity. Since a wireless network continuously changes due to, for ex- ample, power restrictions, sensors faults, sensors disconnection etc., it is needed to adapt the MCDS to the new network configuration. In this paper, we study the influence of removing a node on the MCDS and we propose a localized reconfiguration algorithm to obtain the MCDS of the new network topology. 1. I NTRODUCTION An ad-hoc wireless network is a decentralized type of wireless network characterized by a lack of fixed communication infrastructure, so the selection of which nodes forward data is dynamically making by considering the current network connectivity. Several researchers have proposed to use of a virtual backbone in wireless ad-hoc networks as an alternative to the fixed routing infrastruc- ture in classical wired networks [1, 2, 5, 7, 10]. The virtual backbone represents the “skeleton” of the entire network and is used to frequency exchange routing information (traffic conditions, neighbourhood information, etc.) and broadcast a message from one node to all the nodes in the networks. A Connected Dominating Set (CDS) is a subset of nodes such that: • any two nodes are joined by a path in the network and • any node in the network either belongs to the CDS (CDS node) or has a neighbour in the CDS (non-CDS node). The Minimum CDS (MCDS) is a natural candidate to be the virtual backbone infrastructure in wireless ad-hoc networks because it guarantees the connectivity of the entire network using the Publication co-financed by the European Union as part of the European Social Fund within the project Center for Applications of Mathematics and by Xunta de Galicia (Spain) under grant with number 2014/037.