# The G-Convexity and the G-Centroids of Composite Graphs
## 2 Definitions and Preliminary Results  
In this section, we present some of the important definitions that are needed to understand the rest of this paper. Standard graph-theoretic definitions that are not given here; the reader may refer to the work in [5]. We now present a recent application of the g-convexity. A Mobile Ad hoc Network (MANET) is a decentralized type of wireless network. It is ad hoc in nature because it does not rely on any preexisting infrastructure such as an Access Point (AP) or a router to route packets from a source to a destination node. The network is formed in fly and mobile nodes may join and leave the network as they wish. Thus, there is no admission or access control mechanism in a MANET. Due to this nature, MANETs are deployed during emergency response operations. Wireless nodes that are within each other's radio range communicate directly like a point-to-point network. There is no switch to facilitate communication as a wired network. Nodes that are not within the radio range of each other can still communicate, provided that intermediate nodes act as a router to rely on their packets. Thus, the network is formed through cooperation from all other nodes in the network. Due to this cooperative process, nodes lose their energy on forwarding packets on behalf of other nodes. This is in addition to their transmission and reception. A mobile node spends more than 60% of its power on transmission and reception compared with its internal processing. Thus, energy conservation is one of the important problems in MANET. Therefore, every node needs to know how many pairs of nodes that require its help in forwarding a data packet. Several routing protocols designed for MANET uses some form of the shortest path algorithm to forward packets between a pair of nodes. Thus, if a mobile node \(z\) lies in the shortest path between two other nodes \(u\) and \(v\), then \(z\) is expected to route packets for \(u\) and \(v\). The following is one of the important problems in MANETs. For a mobile node \(u\), locating the maximum set \(S\) of nodes that can communicate with each other without the help of \(u\). WiFi interface is known to be a primary energy consumption in mobile devices. The \"idle listening\" consumes more energy compared with transmission or reception. It is estimated that about 60% of the energy is wasted in idle listening. The only solution for reducing idle listening is to implement a sleep schedule. Thus, intuitively whenever the nodes in \(S\) are communicating, \(u\) can go to sleep mode to conserve its battery power from idle listening. MANET can be modeled as a simple undirected graph. An edge between the mobile hosts \(u\) and \(v\) indicates that both \(u\) and \(v\) are within their radio ranges. To simplify our discussion, we assume that if \(u\) is within the transmission range of \(v\), then \(v\) will be in the transmission range of \(u\) (i.e., the relation between the nodes are symmetric. This may not be true in general due to different factors like the transmission power of a node, geographical conditions, etc. ; thus, the resulting graph is a directed graph). Thus, the resulting graph is undirected. We formally define the g-convexity and various parameters that are defined through the g-convexity. **Definition 1**. : _A set \(S\subseteq V\) is geodetic convex (g-convex for short) if for every pair of vertices \(u,v\in S\), all vertices on any \(u-v\) shortest path (also called a geodesic path) belong to \(S\)._  
From the above definition, it easily follows that a singleton set, vertex pair of an edge, and the whole vertex set \(V(G)\) are g-convex sets of \(G\). We call them as trivial g-convex sets. Moreover, if \(S\) is a clique (\(S\) induces a complete subgraph of \(G\)), then \(S\) is a g-convex set of \(G\). A convex set is a set of vertices which is \"closed\" for the flow of information (routing, control, or data packets). This is in line with [PERSON]'s treatment of Interval functions in a graph [6].