Affidavit of Assumption 2

 20 August 01:27   

    A blueprint is a backwoods if and alone if for every brace of audible vertices $u,v$, there is at alotof one $u,v$-path.

    Suppose you accept a blueprint $G$ which is not a forest. Back a backwoods is to be a blueprint absolute no cycles, it is bright that $G$ haveto accept a aeon $C$. All cycles accept at atomic three vertices. Let $v\_1,\; v\_2,\; ldots,\; v\_n$ (with $n/ge\; 3$) be the vertices on $C$ in order. Then there are two audible $v\_1,v\_2$-paths, namely $v\_1v\_2$ and $v\_1v\_nv\_ldots\; v\_3v\_2$.

    Now accept that $G$ is not a forest, but has two audible $u,v$-paths, namely $ua\_1a\_2ldots\; a\_nv$ and $ub\_1b\_2ldots\; b\_mv$. We would like to say that these make a aeon $ua\_1ldots\; a\_nvb\_mb\_ldots\; b\_1u$. However, a aeon can alone cover anniversary acme once, and it is accessible that one of the $b\_i$ is according to one of the $a\_j$. To that end, address $a\_0=b\_0=u$ and $a\_=b\_=v$. Let $i$ be the better amount of such that $a\_i=b\_j$ for some $j$. Then we acquisition a cycle, namely $a\_ia\_ldots\; a\_nvb\_mb\_ldots\; b\_j$. By our best of $i$, acutely no acme appears added than once.