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

1 Basic definitions

Let $G = (V, E)$ be a $2$ -edge-connected undirected graph of $n$ vertices. Given an edge $e \in E$ , we denote by $G - e = (V, E ∖ {e})$ the graph obtained after the removal of $e$ from $G$ . Let $T$ be a tree spanning $V$ which is also a subgraph of $G$ . Given an edge $e$ of $T$ , let $S_{e}$ be the set of all the swap edges for $e$ , i.e., all edges in $E ∖ {e}$ whose endpoints lie in two different connected components of $T - e$ . For any edge $e$ of $T$ and $f \in S_{e}$ , let $T_{e / f}$ denote the swap tree obtained from $T$ by replacing $e$ with $f$ . Given two vertices $x, y \in V$ , we denote by $d_{G} (x, y)$ the distance between $x$ and $y$ in $G$ , i.e., the number of edges contained in a shortest path in $G$ between $x$ and $y$ . We define the stretch factor $σ_{G} (T)$ of $T$ w.r.t. $G$ as

σ_{G} (T) = max x, y \in V \frac{d_{T} (x, y)}{d_{G} (x, y)} .

[Best Swap Edge] Let $e$ be an edge of $T$ . An edge $f^{*} \in S_{e}$ is a best swap edge for $e$ if $f^{*} \in arg {min}_{f \in S_{e}} σ_{G - e} (T_{e / f})$ . For a swap edge $f = (x, y) \in S_{e}$ we say that $g = (a, b) \in S_{e}$ is critical for $f$ if $σ_{S} (T_{e / f}) = d_{T} (x, a) + 1 + d_{T} (b, y)$ . A set $C$ is critical for $e$ if, for every swap edge $f \in S_{e}$ , $C$ contains a critical edge for $f$ .

2 Our result

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.

Let $e$ be any fixed edge of $T$ . Let $X$ be the set of vertices contained in one of the two connected components of $T - e$ (ties are chosen arbitrarily). Let $Y = V (G) ∖ X$ be the vertices contained in the other connected component of $T - e$ . For every $x \in X$ and for every two (not necessarily distinct) edges $g = (a, b), g^{'} = (a^{'}, b^{'}) \in S_{e}$ , with $a, a^{'} \in X$ and $b, b^{'} \in Y$ , we define

ϕ_{x} (g, g^{'}) := d_{T} (x, a) + d_{T} (b, b^{'}) + d_{T} (a^{'}, x) .

We denote by $(g_{x}, g_{x}^{'})$ a pair of (swap) edges in $arg {max}_{(g, g^{'}) \in S (e)^{2}} ϕ_{x} (g, g^{'})$ . We use the notation $g_{x} = (a_{x}, b_{x})$ and $g_{x}^{'} = (a_{x}^{'}, b_{x}^{'})$ , with $a_{x}, a_{x}^{'} \in X$ and $b_{x}, b_{x}^{'} \in Y$ . We denote by $T_{X}$ the subtree of $T$ induced by the vertices in $X$ . Finally, for every edge $e^{'} = (x, y)$ of $T_{X}$ , we denote by $U (e^{'}, x)$ and $U (e^{'}, y)$ the partition of the vertices $X$ induced by $T_{X} - e^{'}$ and containing $x$ and $y$ , respectively. In [1] it is shown that for every $f = (x, y) \in S_{e}$ , a critical edge for $f$ is either $g_{x}$ or $g_{x}^{'}$ .

Let $e^{'} = (x, z) \in E (T_{X})$ such that $ϕ_{z} (g_{x}, g_{x}^{'}) < ϕ_{z} (g_{z}, g_{z}^{'})$ . Then, one of the following two conditions is satisfied:

$a_{z}, a_{z}^{'} \in U (e^{'}, x)$ and ${a_{x}, a_{x}^{'}} \cap U (e^{'}, z) \neq \emptyset$ ;
$a_{x}, a_{x}^{'} \in U (e^{'}, z)$ and ${a_{z}, a_{z}^{'}} \cap U (e^{'}, x) \neq \emptyset$ .

Proof.

We first show that either $a_{z}, a_{z}^{'} \in U (e^{'}, x)$ or $a_{x}, a_{x}^{'} \in U (e^{'}, z)$ (or even both conditions) must hold. For the sake of contradiction, assume this is not the case. W.l.o.g., let $a_{x}, a_{z} \in U (e^{'}, x)$ and $a_{x}^{'}, a_{z}^{'} \in U (e^{'}, z)$ . In this case, $ϕ_{z} (g_{z}, g_{z}^{'}) = ϕ_{x} (g_{z}, g_{z}^{'}) \leq ϕ_{x} (g_{x}, g_{x}^{'}) = ϕ_{z} (g_{z}, g_{z}^{'})$ . Therefore, $ϕ_{z} (g_{z}, g_{z}^{'})$ cannot be stricly smaller that $ϕ_{z} (g_{x}, g_{x}^{'})$ .

Now we show that ${a_{x}, a_{x}^{'}} ⊈ U (e^{'}, z)$ . For the sake of contradiction, assume that $a_{x}, a_{x}^{'} \in U (e^{'}, x)$ . We have that $ϕ_{z} (g_{z}, g_{z}^{'}) \leq ϕ_{x} (g_{z}, g_{z}^{'}) + 2 \leq ϕ_{x} (g_{x}, g_{x}^{'}) + 2 = ϕ_{z} (g_{x}, g_{x}^{'})$ . Therefore, $ϕ_{z} (g_{z}, g_{z}^{'})$ cannot be stricly smaller that $ϕ_{z} (g_{x}, g_{x}^{'})$ in this case.

Finally, we show that ${a_{z}, a_{z}^{'}} ⊈ U (e^{'}, x)$ . For the sake of contradiction, assume that $a_{z}, a_{z}^{'} \in U (e^{'}, z)$ . From $ϕ_{z} (g_{x}, g_{x}^{'}) < ϕ_{z} (g_{z}, g_{z}^{'})$ we derive $ϕ_{x} (g_{x}, g_{x}^{'}) \leq ϕ_{z} (g_{x}, g_{x}^{'}) + 2 < ϕ_{z} (g_{z}, g_{z}^{'}) + 2 = ϕ_{x} (g_{x}, g_{x}^{'})$ , thus contradicting the choice of $g_{x}$ and $g_{x}^{'}$ . The claim follows. ∎

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

Proof.

Let $e$ be any fixed edge of $T$ . Let $e^{'} = (x, z) \in E (T_{X})$ such that $ϕ_{z} (g_{x}, g_{x}^{'}) < ϕ_{z} (g_{z}, g_{z}^{'})$ . If such an edge does not exist, then the critical set of $e$ has size at most 2. Therefore, we assume that such an edge exists. From Lemma 2, one of the following two conditions is satisfied:

$a_{z}, a_{z}^{'} \in U (e^{'}, x)$ and ${a_{x}, a_{x}^{'}} \cap U (e^{'}, z) \neq \emptyset$ ;
$a_{x}, a_{x}^{'} \in U (e^{'}, z)$ and ${a_{z}, a_{z}^{'}} \cap U (e^{'}, x) \neq \emptyset$ .

W.l.o.g., we assume that $a_{z}, a_{z}^{'} \in U (e^{'}, x)$ and ${a_{x}, a_{x}^{'}} \cap U (e^{'}, z) \neq \emptyset$ . We prove that for every $z^{'} \in U (e^{'}, z)$ , ${g_{z^{'}}, g_{z^{'}}^{'}} = {g_{z}, g_{z}^{'}}$ . Indeed, $ϕ_{z^{'}} (g_{z^{'}}, g_{z^{'}}^{'}) \leq ϕ_{z} (g_{z^{'}}, g_{z^{'}}^{'}) + 2 d_{T} (z, z^{'}) \leq ϕ_{z} (g_{z}, g_{z}^{'}) + 2 d_{T} (z, z^{'}) = ϕ_{z^{'}} (g_{z}, g_{z}^{'})$ . As a consequence, if for every vertex $x^{'} \in U (e^{'}, x)$ , ${g_{x^{'}}, g_{x^{'}}^{'}} = {g_{x}, g_{x}^{'}}$ , then ${g_{x}, g_{x}^{'}, g_{z}, g_{z}^{'}}$ would be a critical set of $e$ of size at most 4. Therefore, we only need to prove the claim when there exists a vertex $y \in U (e^{'}, x)$ such that ${g_{y}, g_{y}^{'}} \neq {g_{x}, g_{x}^{'}}$ .

Let $x^{'} \in U (e^{'}, x)$ be the vertex closest to $x$ such that ${g_{x^{'}}, g_{x^{'}}^{'}} = {g_{x}, g_{x}^{'}}$ and there exists a neighbor $y$ of $x^{'}$ in $U (e^{'}, x)$ such that $ϕ_{y} (g_{x}, g_{x}^{'}) < ϕ_{y} (g_{y}, g_{y}^{'})$ . Let $e^{''} = (x^{'}, y)$ . From Lemma 2, one of the following two conditions is satisfied:

$a_{y}, a_{y}^{'} \in U (e^{''}, x^{'})$ and ${a_{x}, a_{x}^{'}} \cap U (e^{''}, y) \neq \emptyset$ ;
$a_{x}, a_{x}^{'} \in U (e^{''}, y)$ and ${a_{y}, a_{y}^{'}} \cap U (e^{''}, x^{'}) \neq \emptyset$ .

Since $U (e^{'}, z) \subseteq U (e^{''}, x^{'})$ , we have that ${a_{x}, a_{x}^{'}} \cap U (e^{''}, x^{'}) \neq \emptyset$ . Therefore, we can exclude the second of the two conditions and claim that $a_{y}, a_{y}^{'} \in U (e^{''}, x^{'})$ as well as ${a_{x}, a_{x}^{'}} \cap U (e^{''}, y) \neq \emptyset$ . We prove that for every $y^{'} \in U (e^{''}, y)$ , ${g_{y^{'}}, g_{y^{'}}^{'}} = {g_{y}, g_{y}^{'}}$ . Indeed, $ϕ_{y^{'}} (g_{y^{'}}, g_{y^{'}}^{'}) \leq ϕ_{y} (g_{y^{'}}, g_{y^{'}}^{'}) + 2 d_{T} (y, y^{'}) \leq ϕ_{y} (g_{y}, g_{y}^{'}) + 2 d_{T} (y, y^{'}) = ϕ_{y^{'}} (g_{y}, g_{y}^{'})$ . As a consequence, if for every vertex $x^{'} \in U (e^{'}, x) \cap U (e^{''}, x^{'})$ , ${g_{x^{'}}, g_{x^{'}}^{'}} = {g_{x}, g_{x}^{'}}$ , then ${g_{x}, g_{x}^{'}, g_{z}, g_{z}^{'}, g_{y}, g_{y}^{'}}$ would be a critical set of $e$ of size at most 6. Therefore, we only need to prove the claim when there exists a vertex $t \in U (e^{'}, x) \cap U (e^{''}, x^{'})$ such that ${g_{t}, g_{t}^{'}} \neq {g_{x}, g_{x}^{'}}$ . We conclude the proof by showing that such a vertex cannot exist. For the sake of contradiction, let $x^{''} \in U (e^{'}, x) \cap U (e^{''}, x^{'})$ be the vertex that minimizes the sum of distances from itself to both $x$ and $x^{'}$ such that ${g_{x^{''}}, g_{x^{''}}^{'}} = {g_{x}, g_{x}^{'}}$ and there exists a neighbor $t$ of $x^{''}$ in $U (e^{'}, x) \cap U (e^{''}, x^{'})$ such that $ϕ_{t} (g_{x}, g_{x}^{'}) < ϕ_{t} (g_{t}, g_{t}^{'})$ . Let $e^{'''} = (x^{''}, t)$ . From Lemma 2, one of the following two conditions is satisfied:

$a_{t}, a_{t}^{'} \in U (e^{'''}, x^{''})$ and ${a_{x}, a_{x}^{'}} \cap U (e^{'''}, t) \neq \emptyset$ ;
$a_{x}, a_{x}^{'} \in U (e^{'''}, t)$ and ${a_{y}, a_{y}^{'}} \cap U (e^{'''}, x^{''}) \neq \emptyset$ .

Since $U (e^{'}, z), U (e^{''}, y) \subseteq U (e^{'''}, x^{''})$ , then $a_{x}, a_{x}^{'} \in U (e^{'''}, x^{''})$ . As a consequence, none of the two conditions can be satisfied. Hence, $t$ 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.

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

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Proof.

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