# What is the difference between a full and a faithful graph homomorphism?

In their paper "Graph homomorphisms: structure and symmetry" Gena Hahn and Claude Tardif introduce the subject of graph homomorphism "in the mixed form of a course and a survey".

Let $G$ and $H$ be [simple] graphs. A function $\phi : V (G) \longrightarrow V (H)$ is a homomorphism from $G$ to $H$ if it preserves edges, that is, if for any edge $[u,v]$ of $G$, $[\phi(u),\phi(v)]$ is an edge of $H$. We write simply $\phi : G \longrightarrow H$.

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A homomorphism $\phi : G \rightarrow H$ is called faithful if $\phi(G)$ is an induced subgraph of $H$. It will be called full if $[u,v] \in E(G)$ if and only if $[\phi(u),\phi(v)] \in E(H)$, that is, when $\phi^{-1}(x) \cup \phi^{-1}(y)$ induces a complete bipartite graph whenever $[x,y] \in E(H)$.

In other words, a homomorphism $\phi : G \rightarrow H$ is faithful when there is an edge between any two pre-images $\phi^{-1}(u)$ and $\phi^{-1}(v)$ such that $[u,v]$ is an edge of $H$. When a faithful homomorphism $\phi$ is bijective, it is full since each $\phi^{-1}(u)$ is a singleton, and we have that $[\phi^{-1}(u),\phi^{-1}(v)]$ is an edge in $G$ if and only if $[u,v]$ is an edge in $H$.

I'm having some difficulties with these definitions, insofar that I don't really get the difference, my intuitive understanding tells me its the same.

I think that a full homomorphism is always faithful, is that correct? Can you give an example of a non full, but faithful homomorphism? Additional imagery would be very kind.

Sidenote: I'm wondering how to generalize these definitions to non simple graphs and am certain that this had been done before, regarding this please see my literature request.

Let $\phi:G\rightarrow H$ be a homomorphism and consider every pair of $X=\phi^{-1}(u)$ and $Y=\phi^{-1}(v)$ where $u$ and $v$ are adjacent vertices of $H$. $\phi$ is faithful if and only if for all such $(X,Y)$, there exist $x\in X$ and $y\in Y$ such that $x$ and $y$ are adjacent. $\phi$ is full if and only if for all such $(X,Y)$, for all $x\in X$ and $y\in Y$, $x$ and $y$ are adjacent. So a full homomorphism is always faithful, but the converse fails.
For example, consider the depicted homomorphism $\phi$ from a graph $G$ of order $6$ to a graph $H$ of order $3$. The circled groups of vertices $X$, $Y$, and $Z$ are the pre-images of the vertices $u$, $v$, and $w$, respectively. The edges of $H$ are $uv$ and $vw$, so because $G$ has edges between the corresponding pre-image pairs of $(X,Y)$ and $(Y,Z)$, $\phi$ is faithful. For $\phi$ to be full, these pairs should induce complete bipartite graphs. This is true for the pair $(X,Y)$, but not for $(Y,Z)$ because $yz_2$ is not an edge of $G$, so $\phi$ is not full. 