Consider the three transactions T$1,$ T$2,$ and T$3,$ and the schedules S$1$ and S$2$ given below.

T$1$: r$1($x$)$; r$1($z$)$; w$1($x$)$

T$2$: r$2($z$)$; r$2($y$)$; w$2($z$)$; w$2($y$)$

T$3$: r$3($x$)$; r$3($y$)$; w$3($y$)$

S$1$: r$1($x$)$; r$2($z$)$; r$1($z$)$; r$3($x$)$; r$3($y$)$; w$1($x$)$; c$1$; w$3($y$)$; r$2($y$)$; c$3$; w$2($z$)$; w$2($y$)$; c$2$

S$2$: r$1($x$)$; r$2($z$)$; r$1($z$)$; r$3($x$)$; r$3($y$)$; w$1($x$)$; r$2($y$)$; w$3($y$)$; w$2($z$)$; w$2($y$)$; c$1$; c$2$; c$3$

Notes: r$1($x$)$ represents transaction $1$ read x; w$2($y$)$ represents transaction $2$ write y; c$3$ represents transaction $3$ commits.

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$1.$ Draw the serializability graphs for S$1$ and S$2$ and state whether each schedule is serializable or not. If a schedule is serializable, write down the equivalent serial schedule$($s$).[20$ marks$]$

$2.$Determine whether the schedules S$1$ and S$2$ are strict, cascadeless, recoverable, or nonrecoverable. $[20$ marks$]$