Projections in Graph Products I was going throug the topic Projections in Graph Products. There a term weak homomorphism was used. Although I am familiar with the term Isomorphism of graphs, but have no idea what this term "weak homomorphism" means here.
PROJECTION: Let * represent either cartesian, the direct or strong product of graphs, and consider a product $G_1 * G_2* ....*G_k$. For any index i 1$\leq$*i*$\leq$*k*, a Projection map is defined as :
$p_i$ : $G_1 * G_2* ....*G_k$ $\rightarrow$ $G_i$ where $p_i$($x_1,x_2,...,x_k$)=$x_i$.
Can anybody help me out with the term weak homomorphism.
Thanks a lot.
Following links will help for definiton of product graphs...
http://en.wikipedia.org/wiki/Tensor_product_of_graphs
http://en.wikipedia.org/wiki/Cartesian_product_of_graphs
http://en.wikipedia.org/wiki/Strong_product_of_graphs
 A: (We don’t actually need to know about the graph products in order to answer the question.)
If $G$ and $H$ are graphs with vertex sets $V(G)$ and $V(H)$, respectively, a homomorphism from $G$ to $H$ is a map $\varphi:V(G)\to V(H)$ that preserves adjacency: if $uv$ is an edge in $G$, then $\varphi(u)\varphi(v)$ is an edge in $H$. A weak homomorphism from $G$ to $H$ is a map $\varphi:V(G)\to V(H)$ such that if $uv$ is an edge in $G$, then either $\varphi(u)\varphi(v)$ is an edge in $H$, or $\varphi(u)=\varphi(v)$. In other words, if $\varphi$ is a weak homomorphism from $G$ to $H$, and $e$ is an edge in $G$, $\varphi$ either preserves $e$ as an edge by collapses it to a single vertex. Clearly every homomorphism is automatically a weak homomorphism.
In the diagram below let $G$ be the path graph on the left and $H$ the triangle graph on the right.
           1---2---3                a---b  
                                     \ /  
                                      c

The map 
$$\left\{\begin{align*}
1&\mapsto a\\
2&\mapsto b\\
3&\mapsto c
\end{align*}\right.$$
is a homomorphism from $G$ to $H$ is a homomorphism. The map
$$\left\{\begin{align*}
1&\mapsto a\\
2&\mapsto b\\
3&\mapsto b
\end{align*}\right.$$
is not a homomorphism, because it does not take the edge $23$ of $G$ to an edge of $H$, but it is a weak homomorphism: it collapses that edge to the single vertex $b$ of $H$.
