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1. Node n ′ {\displaystyle n'} has received broadcast message but it has not sent a convergecast message back to the core. In this case, fragment F {\displaystyle F} can simply join the broadcast process of F ′ {\displaystyle F'} . Specifically, we image F {\displaystyle F} and F ′ {\displaystyle F'} have already combined to form a new fragment F ″ {\displaystyle F''} , so we want to find the minimum weight outgoing edge of F ″ {\displaystyle F''} . In order to do that, node n ′ {\displaystyle n'} can initiate a broadcast to F {\displaystyle F} to update the fragment ID of each node in F {\displaystyle F} and collect minimum weight outgoing edge in F {\displaystyle F} .

2. Node n ′ {\displaystyle n'} has already sent a convergecast message back to the core. Before node n ′ {\displaystyle n'} sent a convergecast message, it must have picked a minimum weight outgoing edge. As we discussed above, n ′ {\displaystyle n'} does that by choosing its minimum weight basic edge, sending a test message to the other side of the chosen edge, and waiting for the response. Suppose e ′ {\displaystyle e'} is the chosen edge, we can conclude the following: The second statement follows if the first one holds. For the first statement, suppose n ′ {\displaystyle n'} chose the edge e {\displaystyle e} and sent a test message to n {\displaystyle n} via edge e {\displaystyle e} . Then, node n {\displaystyle n} will delay the response (according to case 3 of "Finding the minimum weight incident outgoing edge"). Then, it is impossible that n ′ {\displaystyle n'} has already sent its convergecast message. By the aforementioned conclusions 1 and 2, we can conclude it is safe to absorb F {\displaystyle F} into F ′ {\displaystyle F'} since e ′ {\displaystyle e'} is still the minimum outgoing edge to report after F {\displaystyle F} is absorbed.

1. e ′ ≠ ≠ e {\displaystyle e'\neq e}

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