# The G-Convexity and the G-Centroids of Composite Graphs
## 3 Graph Compositions
### The Join of Graphs
Thus, \(w(x)=n_{1}-1-\omega(G_{2})\), where \(n_{1}=\mid\(G_{2}\), not containing \(y\). Then, \(M\cup V(G_{1})\) is a g-convex set of \(G\) not containing \(y\) and \(|\ M\cup V(G_{1})\ |\ \geq\omega(G_{2})-1+n_{1}=w(x)\). Thus, \(w(y)\geq w(x)\). Since \(y\) is arbitrary, we have \(V(G_{1})\subseteq C_{g}(G)\). Corollary 3: _Let \(G_{1}\) be a complete graph and \(G_{2}\) be an incomplete graph and \(G=G_{1}\lor G_{2}\). Let \(N_{1}\),\(N_{2}\), \(\cdots\), \(N_{s}\) be the maximum cliques of \(G_{2}\) and \(N=\cap_{j=1}^{s}N_{j}\). If \(N=\emptyset\), then \(C_{g}(G)=V(G_{1})\)._  
Proof: If \(x\in V(G_{1})\), then by the proof of the above Proposition, \(w(x)=n_{1}+\omega(G_{2})-1\). Let \(y\in V(G_{2})\). Since \(N=\emptyset\), there exists an \(i\), \(1\leq i\leq s\) such that \(y\
ot\in N_{i}\). Then, \(N_{i}\cup V(G_{1})\) is a g-convex set of \(G\) not containing \(y\) with cardinality \(\omega(G_{2})+n_{1}\). Thus, \(w(y)>w(x)\), and therefore \(C_{g}(G)=V(G_{1})\). Next, we describe the structure of \(C_{g}(G)\) under this case. Proposition 18: _Let \(G=G_{1}\lor G_{2}\) with \(G_{1}\) complete and \(G_{2}\) incomplete. Then \(C_{g}(G)\) induces a complete subgraph of \(G\)._  
Proof: From Proposition 17, \(V(G_{1})\subseteq C_{g}(G)\). If \(V(G_{1})=C_{g}(G)\), then \(<C_{g}(G)>\) is complete. If \(|\ C_{g}(G)\cap V(G_{2})\ |=1\), then also \(<C_{g}(G)>\) is complete. Let \(|\ A=C_{g}(G)\cap V(G_{2})\ |\geq 2\). Let \(u,v\in A\). We now show that \(uv\in E(G)\). As \(w(x)=\omega(G_{2})+n_{1}-1\), for each vertex \(x\) of \(G_{1}\), we have \(w(u)=w(v)\) = \(\omega(G_{2})+n_{1}-1\) (as \(V(G_{1})\subseteq C_{g}(G)\)). If \(M\) is any maximum clique of \(G_{2}\), then \(u,v\in M\) (otherwise, \(M\cup V(G_{1})\) is a g-convex set of \(G\) not containing them of cardinality \(\omega(G_{2})+n_{1}\)). As \(M\) is a clique, \(uv\in E(G_{2})\), and hence \(u\) and \(v\) are adjacent in \(G\) also. This proves that \(<C_{g}(G)>\) is complete. The proof for the following corollary is immediate from Proposition 18. Corollary 4: _Let \(G_{1}\) be a complete graph and \(G_{2}\) be an incomplete graph and \(G=G_{1}\lor G_{2}\). Let \(N_{1}\),\(N_{2}\), \(\cdots\), \(N_{s}\) be the maximum cliques of \(G_{2}\). If \(C_{g}(G)\cap V(G_{2})\
eq\emptyset\), then \(C_{g}(G)\cap V(G_{2})\subseteq\cap_{i=1}^{s}N_{i}\)._  
The containment can be proper. This is demonstrated in Figure 2.  
Proposition 19: _Let \(G_{1}\) be a complete graph and \(G_{2}\) be an incomplete graph and \(G=G_{1}\lor G_{2}\). If \(u\in C_{g}(G)\cap V(G_{2})\), then \(e_{G_{2}}(u)\leq 3\)._  
Proof: From the above corollary, \(u\in\cap_{i=1}^{s}N_{i}\). Thus all the maximum cliques of \(G_{2}\) are contained in \(N_{G_{2}}[u]\). Suppose that \(e_{G_{2}}(u)\geq 4\). Let \(x\in N_{4}(u:G_{2})\). Then \((N_{i}-u)\cup\{x\}\cup V(G_{1})\) is a convex set of \(G\) (as any \(x-y\) geodesic path passes through only vertices from \(V(G_{1})\forall y\in(N_{i}-x)\)), \(x-y\) geodesic contains vertices only from \(V(G_{1})\) with cardinality \(\omega(G_{2})+n_{1}\). This is a contradiction. This proves that \(e_{G_{2}}(u)\leq 3\).