How to represent argmax function for this graph?

Question

I have an input graph which looks like below.

I am trying to find a partial injective function for the above graph. So by definition of partial injective function, I can have unmapped nodes in my left side (for example A5). And by definition of injective function, I look for a one to one function that preserves distinctness. So some possible outputs can be,

Now, I am trying to find just one graph with the maximum weight. How should I represent it in mathematical terms? To be more specific, I am trying to find argmax function or something of that sort.

Answer 1

I think you need to define the set of all possible choice before using argmax. If your graph $G$ has two parts $A = \{A_1, \ldots, A_n\}$ and $L = \{L_1, \ldots, L_m\}$ and arc set $E$, then let

$$D = \{M | M \subseteq E, \forall i \neq j, (A_i, L_k) \in M \Rightarrow (A_j,L_k) \notin M\}$$

Then what you are looking for is

$$ \arg\max_{D' \in D} \sum_{d \in D'} w(d) $$

where $w(d)$ is the weight of arc $d$.