On minimum covering of invertible honeycomb meshes

The problem of monitoring a network by placing a minimum number of sensor devices in the system is modelled as the vertex covering problem (VCP) in graphs. A set S of vertices of a graph G = (V, E) is called a vertex cover, if each edge in E has at least one end point in S and the minimum cardinality taken over all vertex covering sets of G is called the vertex covering number denoted by β(G). This concept has also wide applications in wireless sensor networks and in routing and fault tolerance algorithms. This paper presents the exact values of the vertex covering, edge covering and inverse covering numbers of a popular mesh-derived parallel architecture called the honeycomb mesh network. In particular, we present a characterization for invertible graphs and have shown its significance in electrical networks. In addition, a polynomial time algorithm is provided to find the minimum covering sets in honeycomb meshes.