Complex query and getting results by slug instead of ID in laravel

I have complex query and relation which I'm not fully understand. I'm kind of new in Laravel. Anyway, I'm looking for a way to load this with slugs instead of ID's.

This is the function in the controller

public function index( $category_id)
{
    $Category = new Category;
    $allCategories = $Category->getCategories();
    $category = Category::find($category_id);

    if($category->parent_id == 0) {

         $ids = Category::select('id')->where('parent_id', $category_id)->where('parent_id','!=',0)->get();
         $array = array();

         foreach ($ids as $id) {
            $array[] = (int) $id->id;
         }
         $items = Item::whereIn('category_id',$array)->where('published', 1)->paginate(5);

    } else {
        $items =  Item::where('category_id' ,$category_id)->where('published', 1)->paginate(5);
    }

    return view('list', compact('allCategories','items'));
}

Those are relations in the Model

public function item()
{
    return $this->hasMany('App\Item','category_id');
}

public function children()
{
    return $this->hasMany('App\Category', 'parent_id');
}


public function getCategories()
{
    $categoires = Category::where('parent_id',0)->get();
    $categoires = $this->addRelation($categoires);
    return $categoires;
}
public function selectChild( $id )
{
    $categoires = Category::where('parent_id',$id)->where('published', 1)->paginate(40);
    $categoires = $this->addRelation($categoires);
    return $categoires;
}

public function addRelation( $categoires )
{

  $categoires->map(function( $item, $key)
  {             
        $sub = $this->selectChild($item->id);
        $item->itemCount = $this->getItemCount($item->id , $item->parent_id );
        return $item = array_add($item, 'subCategory', $sub);
    });
    return $categoires;
}
public function getItemCount( $category_id )
{
    return Item::where('category_id', $category_id)->count();
} 

This is what I have in my routes

Route::get('list/{category}', 'ListController@index')->name('list');

currently is loading urls like http://ift.tt/2yBPPeK where 1 is the ID. I'm wonder if with current setup is possible to make it like, http://example.com/slug

I'm aware how slugs are working. I just can't understand how to use them in queries instead of ID's

via Chebli Mohamed