You can have multiple templates for views like :
views-view--home-slideshow-mobile.tpl.php
if your display is a block, you can write it like this :
views-view--home-slideshow-mobile--block.tpl.php
then you have templates for views fields :
views-view-fields--home-slideshow-mobile.tpl.php
Templates for views format (like unformatted for example) :
views-view-unformatted--home-slideshow-mobile.tpl.php
If you want to alter the template of your block, you can declare a block.tpl.php
like :
block--my-block-module--my-block-machinename.tpl.php
You can check all views template in the module views (in like sites/all/contrib/views/theme/...
). You can copy the template and paste them in your theme and rename it to target your specific view.