Notice how every postcode in Leeds has a dominant area? I think the dominant areas are:

LS1/2: City Centre
LS3: Woodhouse
LS4: Burley
LS5: Kirkstall
LS6: Headingley
LS7: Chapel Allerton
LS8: Roundhay
LS9: Gipton
LS10: Middleton
LS11: Beeston
LS12: Armley
LS13: Bramley
LS14: Seacroft
LS15: Crossgates
LS16: Cookridge
LS17: Moortown
LS18: Horsforth
LS19: Yeadon
LS20: Guiseley
LS21: Otley
LS22: Wetherby
LS23: Boston Spa
LS24: Tadcaster
LS25: Garforth
LS26: Rothwell
LS27: Morley
LS28: Pudsey
LS29: Ilkley

As for the districts, I think there are dominant areas in the inner and outer parts.

INNER

East: Seacroft
North East: Roundhay
North West: Headingley
South: Beeston/Middleton
West: Bramley

OUTER

East: Garforth
North East: Alwoodley
North West: Horsforth
South: Morley
West: Pudsey

I'd say the definition of LS3 was incorrect. The vast majority of Woodhouse, and what people consider to be Woodhouse, is in LS6 2, as well as parts of LS2 around the university. I lived in LS3 and it is probably one of the smallest and most insignificant postcodes in Leeds - it's absolutely tiny and consists of the small area between the university and Burley Road, from the inner ring road by the LGI along to Woodsley Road, where LS6 and LS4 begin. It does cover an area called 'Little Woodhouse', although nobody really refers to that very much.

Woodhouse itself is mainly LS6.

ahh... and I'd say LS9 was more Burmantofts than Gipton - lots of Gipton is in LS8.