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In my article, I would like to write the following sentence:

There exisits a vertex in $H$, without loss of generality, let it be $v$, such that $H-v$ is connected.

I think the sentence above is incorrect as two sentences connected by a comma. So I changed it to：

There is a vertex in $H$; let it be $v$ for the sake of generality, such that $H-v$ is connected.

But I feel that the sentence is not fluent. How can I make my sentence more fluent? I try this:

There is a vertex in $H$, namely $v$, such that $H-v$ is connected.

But this does not seem to reflect the meaning of "without loss of generality".

Edits:

PrincessEev's advice is nice. But if I had named $V(H)$ in advance and then wanted to choose $w_1$ without loss of generality, it would have seemed abrupt for me to write that below.

Let $V(H)$ be $\{v_1,v_2,\cdots,v_n\}$. There exists $v_1∈V(H)$ such that...

Or:

"There is a vertex (call it $v_1$) in $H$ such that..."

licheng
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    I would just say "There is a vertex (call it $v$) in $H$ such that..." or "There exists $v \in V(H)$ such that...", myself. This isn't really a "without loss of generality" situation as much as just giving a name to an entity. – PrincessEev Dec 25 '22 at 04:18
  • Thanks. In my article, I have given the names of the vertices of $H$ in advance: $ {v_1,v_2,\cdots, v_n}$, is it OK if I write this: There is a vertex (call it $v_1$) in $H$ such that... ? (The status of each vertices in $v_1,v_2,\cdots, v_n$ is the same.) So I say “without loss of generality... $v_1$” in order to continue the discussion of other vertices ($v_2$, $v_3$, ...) later. – licheng Dec 25 '22 at 04:34
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    In that case, absent any other context, I believe that it would be fine. I would write "There is a vertex in $H$ (without loss of generality, let it be $v_1$) such that..." myself in that scenario. – PrincessEev Dec 25 '22 at 04:42
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    You could also say. "Renumbering if necessary, we can assume <condition on $v_1$>." Of course, assuming that we somehow know that at least one of the $v_i$ satisfy that condition. – JonathanZ Dec 25 '22 at 04:47
  • If you're previously defined $H(V)={v_1,v_2,...,v_n}$ would write it this way "There exists a vertex $v \in H(V)$ such that $H-v$ is connected. Without loss of generality choose $v_1$." If you haven't defined $H(V)$ yet I would simply say "There exists a $v \in H(V)$ such that $H-v$ is connected" and simply drop the second sentence. – CyclotomicField Dec 25 '22 at 11:06

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