SGT

1

Initial flow network:

graph LR;
	s-->|"(8)"|a;
	s-->|"(3)"|b;
	b-->|"(2)"|a;
	b-->|"(4)"|d;
	a-->|"(7)"|p;
	p-->|"(5)"|d;
	p-->|"(3)"|t;
	d-->|"(5)"|t;
	b-->|"(2)"|c;
	c-->|"(1)"|d;
	c-->|"(3)"|h;
	h-->|"(1)"|d;
	h-->|"(3)"|t;

The residual network is the same.

s=>a=>p=>d=>t, bottleneck 5

New flow network:

graph LR;
	s-->|"5(8)"|a;
	s-->|"(3)"|b;
	b-->|"(2)"|a;
	b-->|"(4)"|d;
	a-->|"5(7)"|p;
	p-->|"5(5)"|d;
	p-->|"(3)"|t;
	d-->|"5(5)"|t;
	b-->|"(2)"|c;
	c-->|"(1)"|d;
	c-->|"(3)"|h;
	h-->|"(1)"|d;
	h-->|"(3)"|t;

Residual network:

graph LR;
	s-->|"(3)"|a;
	a-->|"(5)"|s;
	s-->|"(3)"|b;
	b-->|"(2)"|a;
	b-->|"(4)"|d;
	a-->|"(2)"|p;
	p-->|"(5)"|a;
	d-->|"(5)"|p;
	p-->|"(3)"|t;
	t-->|"(5)"|d;
	b-->|"(2)"|c;
	c-->|"(1)"|d;
	c-->|"(3)"|h;
	h-->|"(1)"|d;
	h-->|"(3)"|t;

s=>b=>d=>p=>t, bottleneck 3

New flow network:

graph LR;
	s-->|"5(8)"|a;
	s-->|"3(3)"|b;
	b-->|"(2)"|a;
	b-->|"3(4)"|d;
	a-->|"5(7)"|p;
	p-->|"2(5)"|d;
	p-->|"3(3)"|t;
	d-->|"5(5)"|t;
	b-->|"(2)"|c;
	c-->|"(1)"|d;
	c-->|"(3)"|h;
	h-->|"(1)"|d;
	h-->|"(3)"|t;

Residual network:

graph LR;
	s-->|"(3)"|a;
	a-->|"(5)"|s;
	b-->|"(3)"|s;
	b-->|"(2)"|a;
	b-->|"(1)"|d;
	d-->|"(3)"|b;
	p-->|"(5)"|a;
	a-->|"(2)"|p;
	d-->|"(2)"|p;
	p-->|"(3)"|d;
	t-->|"(3)"|p;
	t-->|"(5)"|d;
	b-->|"(2)"|c;
	c-->|"(1)"|d;
	c-->|"(3)"|h;
	h-->|"(1)"|d;
	h-->|"(3)"|t;

(might be wrong)

Final answer:

graph LR;
	s-->|"7(8)"|a;
	s-->|"3(3)"|b;
	b-->|"(2)"|a;
	b-->|"1(4)"|d;
	a-->|"7(7)"|p;
	p-->|"4(5)"|d;
	p-->|"3(3)"|t;
	d-->|"5(5)"|t;
	b-->|"2(2)"|c;
	c-->|"(1)"|d;
	c-->|"2(3)"|h;
	h-->|"(1)"|d;
	h-->|"2(3)"|t;

2

3

$(s,a)$ with net $3-2+7=8$ $(s,a,b,p,d)$

4

5

6

We know there are two minimum cuts. To increase max. flow by $1$, we must increase the capacity on an edge from each boundary crossing the cuts.