On minimum weakly connected independent sets for wireless sensor networks: properties and enumeration algorithm

Bendali, Fatiha; Mailfert, Jean; Mameri, Djelloul

RAIRO - Operations Research - Recherche Opérationnelle, New challenges in scheduling theory, Tome 49 (2015) no. 2, pp. 313-334.

Modeling topologies in Wireless Sensor Networks principally uses domination theory in graphs. Indeed, many dominating structures have been proposed as virtual backbones for wireless networks. In this paper, we study a dominating set that we call Weakly Connected Independent Set ($wcis$). Given an undirected connected graph $G=(V,E)$, we say that an independent set $S$ in $G$ is weakly connected if the spanning subgraph $(V,[S,V\setminus S\left]\right)$ is connected, where $[S,V\setminus S]$ is the set of edges having exactly one end in $S$. The minimum weakly independent connected set problem consists in determining a $wcis$ of minimum size in $G$. First, we discuss some complexity and approximation results for that problem. Then we propose an implicit enumeration algorithm which computes a minimum $wcis$ in a graph with $n$ vertices with a running time $O^{*}(1.{4655}^n)$ and polynomial space. Processing results are given that show that our enumeration program solves the $mwcis$ problem for graphs whose number of vertices is less than 120.

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