Magento is managed it `Category and product page breadcrumbs`    using `Mage_Catalog_Helper_Data` class function `getBreadcrumbPath` function here magento

check `Category  and product object is exits in current category depends on this magento create breadcrumbs` 

**Note:** 

    You need to write this class

first depend on  `current category  Path` create  an `array` which containing  a `hierarchical category tree ids ` then 

     $pathInStore = $category->getPathInStore();
    $pathIds = array_reverse(explode(',', $pathInStore));

then  create breadcrumbs using below code


                $categories = $category->getParentCategories();

                // add category path breadcrumb
                foreach ($pathIds as $categoryId) {
                    if (isset($categories[$categoryId]) && $categories[$categoryId]->getName()) {
                        $path['category'.$categoryId] = array(
                            'label' => $categories[$categoryId]->getName(),
                            'link' => $this->_isCategoryLink($categoryId) ? $categories[$categoryId]->getUrl() : ''
                        );
                    }
                }

 . 

HOME->ALL CATEGORIES->TOPS->POLO-TOPS.

    As you want to 

TOPS->POLO-TOPS. that means you want only last category then 

you  need modify code if it category page


                $pathInStore = $category->getPathInStore();
                $pathIds = array_reverse(explode(',', $pathInStore));

                $categories = $category->getParentCategories();

                // add category path breadcrumb
		// put All Category id in loop to remove All category from crumns
               foreach ($pathIds as $categoryId) {

		$categoryId= end($pathIds);
                    if (isset($categories[$categoryId]) && $categories[$categoryId]->getName()) {
                        $path['category'.$categoryId] = array(
                            'label' => $categories[$categoryId]->getName(),
                            'link' => $this->_isCategoryLink($categoryId) ? $categories[$categoryId]->getUrl() : ''
                        );
                    }

                }

 And for `remove Home link from catalog` page you need rewrite class 

`Mage_Catalog_Block_Breadcrumbs` and  and remove 

    $breadcrumbsBlock->addCrumb('home', array(
                    'label'=>Mage::helper('catalog')->__('Home'),
                    'title'=>Mage::helper('catalog')->__('Go to Home Page'),
                    'link'=>Mage::getBaseUrl()
                ));
 from rewrite code