**4.4. Conflict graph**

In this subsection the concept of conflict graph is introduced. A conflict graph *Gc*(*Vc, Ec*) consist of the set of edges *Ec* and the set of vertices *Vc*. The vortices *Vc* have a one relation with the set of edges *Ec* of the connectivity graph (i.e. for each edge *e* ∈ *Ec*, there exists a *vc* ∈ *Vc*). As for the set *Ec* of the conflict graph, there exists an edge between two conflict graph vertices *vci* and *vcj* if and only if the corresponding edges *ei* and *ej* of the connectivity graph, are in IE(*e*) set of each other. Hence, if two edges interfere in the connectivity graph, then there is an edge between them in the conflict graph. The conflict graph can now be used to represent any interference model. For instance, we can say that two edges interfere if they use the same wireless channel and they are within interference range. If we want use any other interference model based on signal power, then that can also be easily created by just defining the conditions of interference. Total interference can now be described as the number of links in the conflict graph (i.e. the cardinality of *Ec*).

The above mentioned concepts of connectivity graph, interfering edges and conflict graph are illustrated in Fig.18. For a graph *G*(*V, E*), we find the IE for all the links and then create the conflict graph *Gc*(*Vc, Ec*) (Husnain Mansoor Ali et al., 2009).

two adjacent vertices use the same channel (Husnain Mansoor Ali et al., 2009).

topology are referred to as the *logical links* (Husnain Mansoor Ali et al., 2009).

**4.2. Connectivity graph** 

**4.3. Interfering edges** 

soor Ali et al., 2009).

**4.4. Conflict graph** 

Consider an undirected graph G(V, E) that models the communication network. A graph G, is defined as a set of vertices V and a set of edges E. Each vertex in graph represents a mesh router and each edge between two vertices represents a wireless link between two mesh routers. The color of each vertex represents a non-overlapping channel and the goal of the channel assignment is to cover all vertices with the minimum number of colors such that no

The vertices set *V* consists of the network nodes, which may have multiple radio interfaces (not necessarily the same), while the edges/links set *E* includes all the communication links in the network. A link *e* between a pair of nodes *(vi, vj)*; where *vi, vj* є *V* exists if they are within the communication range of each other and are using the same channel. The graph *G* described above is called the *Connectivity graph* (Fig. 5). The links presented in the network

To include the interference in network model, we introduce the concept of *Interfering edges.*  Interfering edges for an edge *e* (IE(*e*)) are defined as the set of all edges which are using the same channel as edge *e* but cannot use it simultaneously in active state together with edge *e*. All edges are competing for the same channel hence the goal of channel assignment algorithm is to minimize the number of all edges *e* thereby increasing capacity (Husnain Man-

In this subsection the concept of conflict graph is introduced. A conflict graph *Gc*(*Vc, Ec*) consist of the set of edges *Ec* and the set of vertices *Vc*. The vortices *Vc* have a one relation with the set of edges *Ec* of the connectivity graph (i.e. for each edge *e* ∈ *Ec*, there exists a *vc* ∈ *Vc*). As for the set *Ec* of the conflict graph, there exists an edge between two conflict graph vertices *vci* and *vcj* if and only if the corresponding edges *ei* and *ej* of the connectivity graph, are in IE(*e*) set of each other. Hence, if two edges interfere in the connectivity graph, then there is an edge between them in the conflict graph. The conflict graph can now be used to represent any interference model. For instance, we can say that two edges interfere if they use the same wireless channel and they are within interference range. If we want use any other interference model based on signal power, then that can also be easily created by just defining the conditions of interference. Total interference can now be described as the num-

The above mentioned concepts of connectivity graph, interfering edges and conflict graph are illustrated in Fig.18. For a graph *G*(*V, E*), we find the IE for all the links and then create

ber of links in the conflict graph (i.e. the cardinality of *Ec*).

the conflict graph *Gc*(*Vc, Ec*) (Husnain Mansoor Ali et al., 2009).

**Figure 18.** Connectivity graph, interfering edges and conflict graph
