Graphs whose the maximum size of an odd subgraph equal to ⌊n2⌋

A graph is said to be odd if the degrees of all vertices are odd. A subgraph F of X is called an odd factor of X if F is odd and $V\left(F\right)=V\left(X\right)$. Let $f_o^{\prime }\left(X\right)=\max \left\{\right|S|:S\subseteq E(X),X[S\left]\text{ is odd}\right\}$ and $f_o^{″}\left(X\right)=\max \left\{\right|S|:S\subseteq E(X),X[S\left]\text{ is an odd factor of }X\right\}$. In 2001, Scott established that every connected graph X of even order admits a vertex partition $A_1,\dots ,A_s$ such that the induced graph $X\left[A_i\right]$ is odd for $i\in \{1,\dots ,s\}$. It implies that for a graph of order n, $f_o^{\prime }\left(X\right)\ge \lfloor \frac{n}{2}\rfloor$, and $f_o^{″}\left(X\right)\ge \frac{n}{2}$ if n is even. In this paper, first we characterize all trees T with the property that $T-v$ has a perfect matching for any leaf v. Thereby, we comprehensively characterize all connected graphs that attain the tight lower bounds for $f_o^{\prime }\left(X\right)$ and $f_o^{″}\left(X\right)$ respectively.