# The G-Convexity and the G-Centroids of Composite Graphs
## 3 Graph Compositions
### The Join of Graphs
If \(x\in M\), then \((M_{i}-x)\cup N_{j}\), \(1\leq i\leq r\) and \(1\leq j\leq s\) are maximum g-convex sets of \(G\) not containing \(x\), of cardinality \(\omega_{1}+\omega_{2}-1\). Similarly if \(x\in N\), then \(w(x)=\omega_{1}+\omega_{2}-1\). Therefore, the gcws of \(G\) is \(\{(\omega_{1}+\omega_{2}-1)^{m_{1}+m_{2}},(\omega_{1}+\omega_{2})^{n-(m_{1}+m_ {2})}\}\)  
Proofs of (2) and (3) are similar to the proof given above. We now give the structure of \(C_{g}(G)\) under the above cases. If the first case of Proposition 14 is arises, then \(C_{g}(G)=M\cup N\) and hence \(C_{g}(G)\) is a clique of \(G\). In the second case, \(C_{g}(G)\) is either \(M\) or \(N\) depending upon whether \(N\) or \(M\) is empty. Thus in this case also \(C_{g}(G)\) is a clique. In the third case, \(G\) is self centroidal. Thus, the following corollary is obvious. Corollary 2: _If \(G_{1}\) and \(G_{2}\) are incomplete graphs and \(G=G_{1}\lor G_{2}\), then \(C_{g}(G)\) is either a clique of \(G\) or the entire vertex set of \(G\)._  
We now discuss the case when \(G_{1}\) is complete and \(G_{2}\) is incomplete. Here, we show that \(S_{u}(G)=K\cup V(G_{1})\), where \(K\) is a maximum g-convex set in \(G_{2}\) not containing \(u\). Proposition 15: _Let \(G_{1}\) be a complete graph and \(G_{2}\) be an incomplete graph and \(G=G_{1}\lor G_{2}\). Let \(u\in V(G_{1})\). Then \(S_{u}(G)\) induces a complete subgraph of \(G\)._  
The proof follows as in Proposition 13. Proposition 16: _Let \(G_{1}\) be a complete graph and \(G_{2}\) be an incomplete graph and \(G=G_{1}\lor G_{2}\). Let \(u\in V(G_{2})\). Then \(S_{u}(G)=K\cup V(G_{1})\), where \(K\) is a maximum g-convex set in \(G_{2}\) not containing \(u\)._  
Proof: Let \(K^{\prime}\) be a maximum g-convex set of \(G_{2}\) not containing \(u\). We show that \(S=V(G_{1})\cup K^{\prime}\) is a g-convex set of \(G\) not containing \(u\). Let \(x\),\(y\) be a pair of non-adjacent vertices in \(S\). Then, by our definition of \(G\), \(x\),\(y\in K^{\prime}\). If \(d_{G_{2}}(x,y)\geq 3\), then for any \(z\in V(G_{1})\), \(x\),\(z\),\(y\) is a geodesic joining \(x\) and \(y\) in \(G\). Thus, in this case all \(x-y\) geodesics lie in \(<S>\) as \(V(G_{1})\subseteq S\). Suppose that \(d_{G_{2}}(x,y)=2\). Let \(x\),\(z\),\(y\) be any \(x-y\) geodesic in \(G\). If \(z\in V(G_{1})\), then \(z\in S\) as \(V(G_{1})\subseteq S\) otherwise \(z\in K^{\prime}\), as \(K^{\prime}\) is a g-convex set in \(G_{2}\). Thus, in this case also, all \(x-y\) geodesics lie in \(<S>\). Therefore, \(S\) is a g-convex set of \(G\). From the definition of g-weight, it follows that \(w(u\mid G)\geq\mid S\mid\). To prove the equality, it is enough to show that for any \(S_{u}(G)\), \(S_{u}(G)\cap V(G_{2})\) is a g-convex set of \(G_{2}\) not containing \(u\). Let \(K=S_{u}(G)\cap V(G_{2})\). First, we show that \(K\) is non-empty. If \(K\) is empty, then \(S_{u}(G)\subseteq V(G_{1})\). As \(G_{2}\) is incomplete, \(\mid V(G_{2})\mid\geq 3\). Let \(v\) be a vertex of \(G_{2}\) different from \(u\). Then \(S=V(G_{1})\cup\{v\}\) is a g-convex set of \(G\), not containing \(u\), properly containing \(S_{u}(G)\). This is a contradiction. Next we show that \(K\) is a g-convex set of \(G_{2}\). If \(K\) is a clique, then \(K\) is trivially a g-convex set. Let \(x,y\in K\) such that \(d_{G_{2}}(x,y)=2\). Let \(x\),\(z\),\(y\) be a geodesic joining \(x\) and \(y\) in \(G_{2}\). From the definition of \(G\), \(d_{G}(x,y)=2\) and \(x\),\(z\),\(y\) is a geodesic joining \(x\) and \(y\) in \(G\). By the convexity of \(S_{u}(G),z\in S_{u}(G)\), and thus \(z\in K\). Thus \(K\) is a g-convex set of \(G_{2}\). Next we describe the structure of \(C_{g}(G)\) when \(G_{1}\) is complete and \(G_{2}\) is incomplete. Proposition 17: _Let \(G_{1}\) be a complete graph and \(G_{2}\) be an incomplete graph and \(G=G_{1}\lor G_{2}\). \(V(G_{1})\subseteq C_{g}(G)\)._  
Proof: Let \(x\in V(G_{1})\). Then, by Proposition 15, \(S_{x}\) induces a complete subgraph of \(G\).