An Interesting Structural Property Related to the Problem of Computing All the Best Swap Edges of a Tree Spanner in Unweighted Graphs

07/09/2018
by   Davide Bilò, et al.
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In this draft we prove an interesting structural property related to the problem of computing all the best swap edges of a tree spanner in unweighted graphs. Previous papers show that the maximum stretch factor of the tree where a failing edge is temporarily swapped with any other available edge that reconnects the tree depends only on the critical edge. However, in principle, each of the O(n^2) swap edges, where n is the number of vertices of the tree, may have its own critical edge. In this draft we show that there are at most 6 critical edges, i.e., each tree edge e has a critical set of size at most 6 such that, a critical edge of each swap edge of e is contained in the critical set.

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1 Basic definitions

Let be a -edge-connected undirected graph of vertices. Given an edge , we denote by the graph obtained after the removal of from . Let be a tree spanning which is also a subgraph of . Given an edge of , let be the set of all the swap edges for , i.e., all edges in whose endpoints lie in two different connected components of . For any edge of and , let denote the swap tree obtained from by replacing with . Given two vertices , we denote by the distance between and in , i.e., the number of edges contained in a shortest path in between and . We define the stretch factor of w.r.t. as

[Best Swap Edge] Let be an edge of . An edge is a best swap edge for if . For a swap edge we say that is critical for if . A set is critical for if, for every swap edge , contains a critical edge for .

2 Our result

In this draft we show that there are at most 6 critical edges, i.e., each tree edge has a critical set of size at most 6 such that, a critical edge of each swap edge of is contained in the critical set.

Let be any fixed edge of . Let be the set of vertices contained in one of the two connected components of (ties are chosen arbitrarily). Let be the vertices contained in the other connected component of . For every and for every two (not necessarily distinct) edges , with and , we define

We denote by a pair of (swap) edges in . We use the notation and , with and . We denote by the subtree of induced by the vertices in . Finally, for every edge of , we denote by and the partition of the vertices induced by and containing and , respectively. In [1] it is shown that for every , a critical edge for is either or .

Let such that . Then, one of the following two conditions is satisfied:

  • and ;

  • and .

Proof.

We first show that either or (or even both conditions) must hold. For the sake of contradiction, assume this is not the case. W.l.o.g., let and . In this case, . Therefore, cannot be stricly smaller that .

Now we show that . For the sake of contradiction, assume that . We have that . Therefore, cannot be stricly smaller that in this case.

Finally, we show that . For the sake of contradiction, assume that . From we derive , thus contradicting the choice of and . The claim follows. ∎

[6-critical-set theorem] For every edge of , there exists a critical set of having size at most 6.

Proof.

Let be any fixed edge of . Let such that . If such an edge does not exist, then the critical set of has size at most 2. Therefore, we assume that such an edge exists. From Lemma 2, one of the following two conditions is satisfied:

  • and ;

  • and .

W.l.o.g., we assume that and . We prove that for every , . Indeed, . As a consequence, if for every vertex , , then would be a critical set of of size at most 4. Therefore, we only need to prove the claim when there exists a vertex such that .

Let be the vertex closest to such that and there exists a neighbor of in such that . Let . From Lemma 2, one of the following two conditions is satisfied:

  • and ;

  • and .

Since , we have that . Therefore, we can exclude the second of the two conditions and claim that as well as . We prove that for every , . Indeed, . As a consequence, if for every vertex , , then would be a critical set of of size at most 6. Therefore, we only need to prove the claim when there exists a vertex such that . We conclude the proof by showing that such a vertex cannot exist. For the sake of contradiction, let be the vertex that minimizes the sum of distances from itself to both and such that and there exists a neighbor of in such that . Let . From Lemma 2, one of the following two conditions is satisfied:

  • and ;

  • and .

Since , then . As a consequence, none of the two conditions can be satisfied. Hence, does not exist. This completes the proof. ∎

References

  • [1] Davide Bilò, Feliciano Colella, Luciano Gualà, Stefano Leucci, and Guido Proietti. A faster computation of all the best swap edges of a tree spanner. In Christian Scheideler, editor, Structural Information and Communication Complexity - 22nd International Colloquium, SIROCCO 2015, Montserrat, Spain, July 14-16, 2015, Post-Proceedings, volume 9439 of Lecture Notes in Computer Science, pages 239–253. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-25258-2_17.