The Reconstruction Conjecture due to Kelly and Ulam states that every graph with at least 3 vertices is uniquely determined by its multiset of subgraphs $\{G-v: v\in V(G)\}$. Let $diam(G)$ and $\kappa(G)$ denote the diameter and the connectivity of a graph $G$, respectively, and let $\mathcal{G}_2:=\{G: \diam(G)=2\}$ and $\mathcal{G}_3:=\{G:\diam(G)=\diam(\overline{G})=3\}$. It is known that the Reconstruction Conjecture is true if and only if it is true for every 2-connected graph in $\mathcal{G}_2\cup \mathcal{G}_3$. Balakumar and Monikandan showed that the Reconstruction Conjecture holds for every triangle-free graph $G$ in $\mathcal{G}_2\cup \mathcal{G}_3$ with $\kappa(G)=2$. Moreover, they asked whether the result still holds if $\kappa(G)\ge 3$. (If yes, the class of graphs critical for solving the Reconstruction Conjecture is restricted to 2-connected graphs in $\mathcal{G}_2\cup\mathcal{G}_3$ which contain triangles.) In this paper, we give a partial solution to their question by showing that the Reconstruction Conjecture holds for every triangle-free graph $G$ in $\mathcal{G}_3$ and every triangle-free graph $G$ in $\mathcal{G}_2$ with $\kappa(G)=3$. We also prove similar results about the Edge Reconstruction Conjecture.

§ INTRODUCTION

Throughout this paper, we use standard graph theory terminology and notation, as in [12]. Unless stated otherwise, assume $|V(G)|\ge3$ and $|E(G)|\ge4$ for every graph $G$. For vertices $u$ and $v$ in a graph $G$, we denote by $d_G(u,v)$ the length of a shortest path from $u$ to $v$ in $G$. The diameter of a graph $G$, denoted $\diam(G)$, is $\max_{u,v\in V(G)}d_G(u,v)$. We denote by $N_G(v)$ the neighborhood of a vertex $v$ in $G$. For a connected graph $G$, a set $S\subseteq V(G)$ is a cut set-1mm if $G-S$ is disconnected; moreover, if $S=\{v\}$, then $v$ is a cut vertex-3mm. The connectivity of $G$, denoted by $\kappa(G)$2.5mm, is the size of its smallest cut set. For $k\geq2$, a graph $G$ is $k$-connected2mm if its connectivity is at least $k$.

Graph Reconstruction is the study which explores whether a graph can be uniquely determined by its subgraphs. A card of a graph $G$ is a subgraph of $G$ obtained by deleting a single vertex; that is, $G-v$ for some $v\in V(G)$. The multiset $\mathcal{D}(G)$ of cards of $G$ is the deckdeck, $\mathcal{D}(G)$ of $G$, i.e., $\mathcal{D}(G):=\{G-v: v\in V(G)\}$. If $G$ is isomorphic to every graph $H$ with $\mathcal{D}(H)=\mathcal{D}(G)$, then $G$ is reconstructible-2mm.
The most well-studied problem in the area of graph reconstruction is the Reconstruction Conjecture7mm proposed by Ulam [11] and Kelly [6, 7].

For $n\ge 3$, every $n$-vertex graph is reconstructible, i.e., it is uniquely determined by its deck.

This conjecture has attracted a lot of attention. It has been confirmed for certain graph classes such as disconnected graphs, trees, regular graphs, perfect graphs, etc. Although much work has gone into proving the conjecture, it remains widely open even for sparse classes of graphs such as bipartite graphs, planar graphs and graphs of bounded maximum degree. For a detailed survey of results on the Reconstruction Conjecture and graph reconstruction in general, we refer the reader to [8].

Yang proved [13] that the Reconstruction Conjecture is true if and only if it is true for every 2-connected graph. Let $\mathcal{G}_2:=\{G: \diam(G)=2\}$ and $\mathcal{G}_3:=\{G:\diam(G)=\diam(\overline{G})=3\}$$\mathcal{G}_2, \mathcal{G}_3$. Gupta et al. [4] showed that the Reconstruction Conjecture is true if and only if it is true for every graph in $\mathcal{G}_2\cup\mathcal{G}_3$. Combining the above two results, Monikandan and Ramachandran [10] showed that it suffices to consider $2$-connected graphs in $\mathcal{G}_2\cup\mathcal{G}_3$ to prove the Reconstruction Conjecture.

The Reconstruction Conjecture is true if and only if every $2$-connected graph in $\mathcal{G}_2\cup\mathcal{G}_3$ is reconstructible.

As a step towards proving the Reconstruction Conjecture,  Balakumar and Monikandan [1] studied the graphs in Theorem <ref> which are bipartite and those which are triangle-free. They proved the following:

If $G\in\mathcal{G}_2$ and is bipartite, or if $G\in\mathcal{G}_3$ and is 2-connected and bipartite, then $G$ is reconstructible.

If $G\in\mathcal{G}_2\cup\mathcal{G}_3$ and is triangle-free with $\kappa(G)=2$, then $G$ is reconstructible.

Note that Theorem <ref> restricts the graphs in Theorem <ref> to those containing odd cycles. Furthermore, Theorem <ref> gives partial results on the graphs in Theorem <ref> which are triangle-free. Balakumar and Monikandan asked whether the class of graphs in Theorem <ref> could be extended to those with connectivity at least 3. They remarked that a positive answer to their question would restrict the graphs in Theorem <ref> to those containing triangles. Furthermore, narrowing down the classes of graphs critical for proving the Reconstruction Conjecture makes it easier to search for a counterexample, if any. As a partial solution to their question, we prove the following two results.

If $G\in \mathcal{G}_2$ and is triangle-free with $\kappa(G)=3$, then $G$ is reconstructible.

If $G\in\mathcal{G}_3$ and is triangle-free with $\kappa(G)\geq3$, then $G$ is reconstructible.

Observe that this leaves only the open case of every triangle-free graph $G\in\mathcal{G}_2$ with $\kappa(G)\geq4$.

An edge-focused variant of the Reconstruction Conjecture was first proposed by Harary [5]. An edge-cardedge-card/deck $\mathcal{E}\mathcal{D}(G)$-3mm of a graph $G$ is $G-e$ for some $e\in E(G)$, and the edge-deck of $G$ is the multiset $\mathcal{E}\mathcal{D}(G):=\{G-e: e\in E(G)\}$. A graph $G$ is edge-reconstructible3mm if $G$ is isomorphic to every graph $H$ with $\mathcal{E}\mathcal{D}(H)=\mathcal{E}\mathcal{D}(G)$.
Harary [5] proposed the Edge Reconstruction Conjecture12mm which states the following.

Every graph with at least $4$ edges is edge-reconstructible, i.e., it is uniquely determined by its edge-deck.

Greenwell [3] established a connection between the Reconstruction Conjecture and the Edge Reconstruction Conjecture by showing that the deck of $G$ can be recovered from its edge-deck.

If $G$ has at least $4$ edges and no isolated vertices, then $\mathcal{D}(G)$ is uniquely determined by  $\mathcal{E}\mathcal{D}(G)$.

Theorem <ref> implies that a graph $G$ with no isolated vertices and $|E(G)|\ge 4$ is edge-reconstructible if it is reconstructible.

We prove edge-reconstruction analogues of Theorems <ref> and <ref> (in fact, we prove a stronger edge-reconstruction analogue of Theorem <ref>).

If $G\in\mathcal{G}_2$ and is triangle-free, then $G$ is edge-reconstructible.

If $G\in\mathcal{G}_3$ and is triangle-free, then $G$ is edge-reconstructible.

Note that in light of Theorems <ref>, <ref>, and <ref>, to prove Theorem <ref>, it suffices to prove the following.

If $G\in\mathcal{G}_3$ and is triangle-free with $\kappa(G)=1$, then $G$ is reconstructible.

A more general problem is to decide if a graph parameter is uniquely determined by its deck or edge-deck. Given a graph $G$, a graph parameter $p(G)$ is reconstructible (resp. edge-reconstructible) if the value of $p(G)$ is the same for each graph $H$ with $\mathcal{D}(H)=\mathcal{D}(G)$ (resp. $\mathcal{E}\mathcal{D}(H)=\mathcal{E}\mathcal{D}(G)$).
A family of graphs $\mathcal{G}$ is recognizable(edge-) recognizable-3mm if, for each $G\in\mathcal{G}$, every graph $H$ with $\mathcal{D}(H)=\mathcal{D}(G)$ is also in $\mathcal{G}$. Moreover, $\mathcal{G}$ is weakly reconstructibleweakly (edge-) reconstructible1mm if, for each $G\in\mathcal{G}$ and each $H\in\mathcal{G}$ with $\mathcal{D}(H)=\mathcal{D}(G)$, the graph $H$ is isomorphic to $G$. If $\mathcal{G}$ is both recognizable and weakly reconstructible, then it is reconstructible. We define edge-recognizable and weakly edge-reconstructible analogously.
We will need the following results on reconstructible graph parameters and recognizable graph classes.

Given a card $G-v$, the degree of $v$ in $G$, as well as the degrees of the neighbors of $v$ in $G$ are reconstructible. Similarly, given an edge-card $G-e$, the degrees in $G$ of the endpoints of $e$ are edge-reconstructible.

The number of occurrences of any proper subgraph of $G$ is reconstructible.

The connectivity of $G$ is reconstructible.

Both $\mathcal{G}_2$ and $\mathcal{G}_3$ are recognizable.

This paper is organized as follows. In Section <ref>, we prove Theorems <ref>, <ref>, and <ref> which address our results on reconstruction. In Section <ref>, we prove Theorems <ref> and <ref> which address edge reconstruction.

We end this section with the following note. Ideally, we would like an edge reconstruction result similar to Theorem <ref> that would restrict the class of graphs critical for proving the Edge Reconstruction Conjecture to those which lie in $\mathcal{G}_2\cup \mathcal{G}_3$. If such a result is proved, then Theorems <ref> and <ref> would further restrict the class of critical graphs to those containing triangles. We remark that such a result is unlikely to be proved using techniques similar to those used in the proof of Theorem <ref>. In particular, the restricted-diameter result of Gupta et al. [4] relies on the fact that $\mathcal{D}(\overline{G})$ is reconstructible from $\mathcal{D}(G)$ for every graph $G$. This is not true for edge reconstruction. Nevertheless, the advantages of having such a result merit further investigation.

§ RECONSTRUCTION: PROOFS OF THEOREMS <REF>, <REF>, AND <REF>

For every positive integer $k$, denote by $[k]$ the set $\{1,2,\dots,k\}$. Let $G$ be a graph with $\kappa(G)=k\geq3$, and let $S=\{x_1,x_2,\dots,x_k\}$ be a cut set of $G$. For $p\geq2$, denote by $C_1,C_2,\dots,C_p$ the components of $G-S$. A component is trivial if it is a single vertex. We group the vertices in each component into classes based on their neighborhood in $S$. Fix $i\in[p]$ and a vertex $v\in C_i$, and let $\{x_{i_1},x_{i_2},\dots,x_{i_m}\}$ be the neighbors of $v$ in $S$, where $\{i_1,i_2,\dots,i_m\}\subseteq[k]$. We say $v$ is in the class $C_i(\{i_1,i_2,\dots,i_m\})$, where $C_i(\{i_1,i_2,\dots,i_m\})$ denotes the subset of vertices in $C_i$ whose neighbors in $S$ are precisely $\{x_{i_1},x_{i_2},\dots,x_{i_m}\}$. For convenience, we write $C_i(j)$ for $C_i(\{j\})$. We say the class $C_i(\{i_1,i_2,\dots,i_m\})$ contains the index $j$ if $j=i_q$ for some $q\in[m]$. If a vertex $v\in C_i$ is not adjacent to any vertex in $S$, then $v$ is in the class $C_i(\emptyset)$. Similarly, if a vertex $v\in C_i$ is adjacent to every vertex in $S$, then $v$ is in the class $C_i(S)$. Note that vertices which are trivial components are adjacent to every vertex in $S$ (by minimality of $S$), i.e., $C_j=C_j(S)$ for every trivial component $C_j$ of $G-S$. Moreover, in a triangle-free graph, every class of a component forms an independent set, except possibly for $C_i(\emptyset)$.

In their paper [1], Balakumar and Monikandan implicitly introduced the notion of the classes of a component when dealing with triangle-free graphs. This helped give more structure to the graph. Their proofs (for the most part) relied on reconstructing the graph from cards which delete a vertex of degree 1 or a cut vertex. Such vertices are “special" in the sense that their set of neighbors is restricted which narrows down the number of cases to consider. We will use a similar approach. However, we note that the higher the connectivity of the graph is, the less “special" those vertices become. To avoid this problem, we will try to deduce as much about the structure of the graph as possible (using only what we know about its diameter and connectivity) before reconstructing it.

It follows from Lemmas <ref>, <ref>, and <ref> that the class of triangle-free graphs with connectivity $k\in\mathbb{N}$ which lie in $\mathcal{G}_2\cup \mathcal{G}_3$ are recognizable. Therefore, to prove that the graphs in Theorems <ref>, <ref>, and <ref> are reconstructible, it suffices to show that they are weakly reconstructible.

Let $G$ be a triangle-free graph in $\mathcal{G}_2$ with $\kappa(G)=3$. Pick a card $H$ with connectivity 2 and let $H:=G-x_1$ (such a card exists because $G$ has a cut set of size 3 and some card deletes one of its vertices). Let $\{x_2,x_3\}$ be a cut set of size 2 in $H$. Since $\kappa(G)=3$, the set $S:=\{x_1,x_2,x_3\}$ is a cut set of $G$.

Let $C_1,\dots,C_p$ be the components of $G-S$. We show that $G-S$ has at most one nontrivial component. By contradiction, assume $C_1$ and $C_2$ are nontrivial. Consider an edge $u_1u_2$ in $C_1$ and an edge $v_1v_2$ in $C_2$. Since $G$ is triangle-free, at least one of $u_1$ and $u_2$ is adjacent to at most one vertex in $S$; say $u_1$. If $u_1$ is adjacent to no vertex in $S$, then $d_G(u_1,v_1)\geq{3}$ contradicting $\diam(G)=2$. So, by symmetry, assume $u_1$ is only adjacent to $x_1$ in $S$. Note that $v_1$ and $v_2$ are not both adjacent to $x_1$; otherwise, we get a triangle. So, assume $v_1$ is nonadjacent to $x_1$. Now, $d_G(u_1,v_1)\geq3$ contradicting $\diam(G)=2$. Thus, $G-S$ has at most one nontrivial component, say $C_1$.

To reconstruct $G$ from $H$, we need to identify the vertices in $H$ adjacent to $x_1$ in $G$. Observe that each vertex in a trivial component of $G-S$ must be adjacent to each vertex of $S$ in $G$ (in particular, it is adjacent to $x_1$); otherwise, $G$ contains a smaller cut set. Moreover, if $C_1$ is trivial, then $G$ is a complete bipartite graph and is reconstructible by Theorem <ref>. Thus, we assume $C_1$ is nontrivial.

[scale=0.8, every node/.style=scale=0.8]

[thick, rounded corners, fill=gray!30!white] (8.5,1) rectangle (15.5,-9);
[thick] (16.2,-4) node[scale=1.5] $C_1$;

[thick, rounded corners] (4,-3) rectangle (5.5,-6);

[thick] (14.2,-1) edge[bend left] (14.2,-4.4) (14.2,-1) edge[bend left] (14.2,-6.8);
[thick] (9.2,-7.7) edge[out=210, in=-30] (x_3) (9.2,-3.5) – (x_2);
[thick] (C_2) – (x_2) – (C_3) (C_p) – (x_2) (C_2) – (x_3) – (C_3) (C_p) – (x_3);
[thick] (x_3) – (9,-1) – (x_2) (10.5,-0.4) edge[out=30, in=150] (13.3,-0.4);

[scale=2, fill=white, thick](C_2,C_3,C_p)
[fill=black, thick](dot1,dot2,dot3)
[scale=1.5, left](C_2,C_3,C_p)
[scale=2, fill=white, thick](x_2,x_3)
[thick, rounded corners, fill =white] (9,-0.4) rectangle (12,-1.6) (12.4,-0.4) rectangle (14.2,-1.6);
[thick] (10.5,-1) node[scale=1.5] $C_1(\{2,3\})$ (13.3,-1) node[scale=1.5] $C_1(1)$;

[thick, rounded corners, fill =white] (9,-3.8) rectangle (12.4,-5) (12.4,-3.8) rectangle (14.2,-5);
[thick, decorate, decoration=calligraphic brace, amplitude=3mm] (9,-3.7) – (14.2,-3.7) (11.7,-2.9) node[scale=1.2] $C_1(2)\cup C_1(\{1,2\})$;

[thick, rounded corners, fill =white] (9,-6.2) rectangle (12.4,-7.4) (12.4,-6.2) rectangle (14.2,-7.4);
[thick, decorate, decoration=calligraphic brace, mirror, amplitude=3mm] (9,-7.5) – (14.2,-7.5) (11.7,-8.3) node[scale=1.2] $C_1(3)\cup C_1(\{1,3\})$;

[thick] (12,2) node[scale=1.5] $C_1(S)\cup C_1(\emptyset)$;
[thick, ->] (10,2) edge[out=180, in=135] (12,0.5);

[scale=0.8, every node/.style=scale=0.8]

[thick, rounded corners, fill=gray!30!white] (8.5,1) rectangle (15.5,-9);
[thick] (16.2,-4) node[scale=1.5] $C_1$;

[thick, rounded corners] (4,-2) rectangle (5.5,-6);

[thick] (14.2,-1) edge[bend left] (14.2,-4.4) (14.2,-1) edge[bend left] (14.2,-6.8);
[thick] (11.5,-6.2) – (9.8,-5) – (9.8,-6.2) – (11.5,-5);
[thick, dashed] (11.5,-5) – (11.5,-6.2) (9.8,-5) edge[out=-30, in=-150] (11.5,-5) (9.8,-6.2) edge[out=30, in=150] (11.5,-6.2);
[thick] (9.2,-7.7) edge[out=210, in=-30] (x_3) (9.2,-3.5) edge[in=20, out=150] (x_2) (x_1) edge[in=200, out=10] (12.8,-1.6);
[thick] (C_2) – (x_2) – (C_3) (C_p) – (x_2) (C_2) – (x_3) – (C_3) (C_p) – (x_3);
[thick] (x_3) – (9,-1) – (x_2) (C_2) – (x_1) – (C_3) (x_1) – (C_p) (10.5,-0.4) edge[out=30, in=150] (13.3,-0.4);

[scale=2, fill=white, thick](C_2,C_3,C_p)
[fill=black, thick](dot1,dot2,dot3)
[scale=1.5, left](C_2,C_3,C_p)
[scale=2, fill=white, thick](x_1,x_2,x_3)
[thick] (4.75,-1.5) node[scale=1.5] $S$;

[thick, rounded corners, fill =white] (9,-0.4) rectangle (12,-1.6) (12.4,-0.4) rectangle (14.2,-1.6);
[thick] (10.5,-1) node[scale=1.5] $C_1(\{2,3\})$ (13.3,-1) node[scale=1.5] $C_1(1)$;

[thick, rounded corners, fill =white] (9,-3.8) rectangle (10.7,-5) (10.7,-3.8) rectangle (12.4,-5) (12.4,-3.8) rectangle (14.2,-5);
[thick, decorate, decoration=calligraphic brace, amplitude=3mm] (9,-3.7) – (14.2,-3.7) (11.7,-2.9) node[scale=1.2] $C_1(2)\cup C_1(\{1,2\})$;

[thick, rounded corners, fill =white] (9,-6.2) rectangle (10.7,-7.4) (10.7,-6.2) rectangle (12.4,-7.4) (12.4,-6.2) rectangle (14.2,-7.4);
[thick, decorate, decoration=calligraphic brace, mirror, amplitude=3mm] (9,-7.5) – (14.2,-7.5) (11.7,-8.3) node[scale=1.2] $C_1(3)\cup C_1(\{1,3\})$;

[thick] (12,2) node[scale=1.5] $C_1(S)\cup C_1(\emptyset)$;
[thick, ->] (10,2) edge[out=180, in=135] (12,0.5);

The figure shows the structures of $G$ and $H$ in Theorem <ref>. Top: the view in the card $H$. Bottom: the view in $G$. Solid lines between two sets indicate that every vertex in the first set is adjacent to every vertex in the second set. Analogously, dashed lines indicate non-adjacency. The component $C_1$ is divided into classes as defined in Section <ref>. The set $C_1(S)\cup C_1(\emptyset)$ is grayed out to indicate that it is empty.

We identify the classes of $C_1$ with respect to the cut set $S$. Note that $C_1(\emptyset) = \emptyset$; otherwise, pick $v\in C_1(\emptyset)$ and observe that $d_G(v,u)\geq3$ for every $u\in C_i$ ($i\neq1)$, a contradiction. Therefore, $C_1(1)$ consists of all vertices in $C_1$ adjacent to neither $x_2$ nor $x_3$ in $H$. Furthermore, $C_1(S)=\emptyset$; otherwise, pick $v\in C_1(S)$ and let $u\in N_{C_1}(v)$. Since $G$ is triangle-free, $u\in C_1(\emptyset)$, a contradiction. Thus, $C_1(\{2,3\})$ consists of all vertices in $C_1$ adjacent to both $x_2$ and $x_3$ in $H$.

This leaves identifying the vertices of $C_1$ in classes $C_1(2), C_1(3), C_1(\{1,2\})$, and $C_1(\{1,3\})$. Let $A_2:=N_{C_1(2)\cup C_1(\{1,2\})}(C_1(1))$ and $A_3:=N_{C_1(3)\cup C_1(\{1,3\})}(C_1(1))$$A_2,A_3$. Note that every $v\in A_2\cup A_3$ is nonadjacent to $x_1$, since $G$ is triangle-free. So, it suffices to identify the vertices of each of the following “$B$" sets. Let $B_2:=C_1(2)-A_2, B_3:=C_1(3)-A_3, B_{12}:=C_1(\{1,2\})-A_2$, and$B_2,B_3$ $B_{13}:=C_1(\{1,3\})-A_3$; see Figure <ref>$B_{12},B_{13}$. (Observe that $B_{12}=C_1(\{1,2\})$ and $B_{13}=C_1(\{1,3\})$ since $G$ is triangle-free, so no vertex in $C_1(\{1,2\})$ or $C_1(\{1,3\})$ is adjacent to a vertex in $C_1(1)$.)

Recall that, since $G$ is triangle-free, (1) each “$B$" set is independent, and (2) no vertex in $B_{12}$ (resp. $B_{13}$) is adjacent to a vertex in $B_2$ (resp. $B_3$) or a vertex in $B_{13}$ (resp. $B_{12}$). Furthermore, for each $v\in B_2$ and $u\in B_{13}$, there exists no $w\in N_{C_1}(u)\cap N_{C_1}(v)$; otherwise, since $G$ is triangle-free, $w\in C_1(\emptyset)$, a contradiction. Hence, $N_G(u)\cap N_G(v)=\emptyset$ as $N_S(u)\cap N_S(v)=\emptyset$ by definition. Thus, (3) every $v\in B_2$ is adjacent to every $u\in B_{13}$; otherwise, $d_G(u,v)\geq3$, contradicting $\diam(G)=2$. Similarly, (4) every vertex in $B_3$ is adjacent to every vertex in $B_{12}$. Finally, for each $u\in B_2$ and $v\in B_3$, there exists no $w\in N_{C_1}(u)\cap N_{C_1}(v)$; otherwise, $w\in C_1(1)$ and $\{u,v\}\subseteq N_{C_1}(C_1(1))$, contradicting the definitions of $B_2$ and $B_3$. Hence, $N_G(u)\cap N_G(v)=\emptyset$ as $N_S(u)\cap N_S(v)=\emptyset$ by definition. Thus, (5) every $u\in B_2$ is adjacent to every $v\in B_3$; otherwise, $d_G(u,v)\geq3$, contradicting $\diam(G)=2$.

Now let $L_{x_2}:=B_2\cup B_{12}$ and $L_{x_3}:=B_3\cup B_{13}$$L_{x_2},L_{x_3}$. Note that $H[L_{x_2}\cup L_{x_3}]$ is bipartite with parts $L_{x_2}$ and $L_{x_3}$, by (1-2). And, in the card, we are unable to distinguish between $B_2$ and $B_{12}$ in $L_{x_2}$ or $B_3$ and $B_{13}$ in $L_{x_3}$. So, we consider the following cases.

Case 1: $B_{12}$ and $B_{13}$ are both nonempty in $G$. Now, in $H$, there must exist vertices $L_2\subseteq L_{x_2}$ that are not adjacent to every vertex in $L_{x_3}$, and vertices $L_3\subseteq L_{x_3}$ that are not adjacent to every vertex in $L_{x_2}$, by (1-5). This means $L_2$ and $L_3$ are precisely $B_{12}$ and $B_{13}$, respectively. Further, $B_2=L_{x_2}-B_{12}$ and $B_3=L_{x_3}-B_{13}$.

Case 2: $B_{12}$ and $B_{13}$ are both empty in $G$. Now it must be that $d_{C_1}(x_1)=|C_1(1)|$. Note that we can calculate the value of $d_{C_1}(x_1)$ as follows: $d_{C_1}(x_1)=d_G(x_1)-|\cup_{i\neq1}C_i|$, where $d_G(x_1)$ is given by Lemma <ref>. So, in $H$, after identifying the vertices of $C_1(1)$, if $|C_1(1)|=d_{C_1}(x_1)$, then the vertices in $C_1(1)$ are the only neighbors of $x_1$ in $C_1$. This means $B_{12}=B_{13}=\emptyset$, and that $B_2=L_{x_2}$ and $B_3=L_{x_3}$.

Case 3: Neither Case 1 nor Case 2 is true. Let $B_{23}$ be defined analogously to $B_{12}$ and $B_{13}$. Note that by Pigeonhole Principle, at least two of the sets $B_{12}, B_{23}$, and $B_{13}$ are nonempty, or at least two of them are empty. Let $i$ be the index shared by those two sets. If $i=1$, then we are done by Case 1 or Case 2. Otherwise, there exists a card $H'$ that deletes $x_i$ and whose corresponding $B_{ij}$ and $B_{ik}$ sets, where $\{j,k\}=\{1,2,3\}-\{i\}$, are either both empty or both nonempty. Now we can repeat the above arguments for $H'$ (where the roles of $H$ and $H'$, and those of $x_1$ and $x_i$ are interchanged).

Recall that we only need to show that this class of graphs is weakly reconstructible. Let $G$ be a triangle-free graph in $\mathcal{G}_3$ with $\kappa(G)\geq3$, let $S$ be a cut set of $G$ with $|S|=\kappa(G)$, and let $C_1,\dots,C_p$ be the components of $G-S$. Since $\kappa(G)\geq3$, the set $S$ has at least 3 vertices, say $S=\{x_1,x_2,\dots,x_k\}$ where $k\geq3$. Moreover, since $\diam(\overline{G})=3$, there exist vertices $u,v\in V(G)$ such that $d_{\overline{G}}(u,v)=3$. This means $d_G(u,v)=1$; that is, $uv\in E(G)$. Observe that $d_{\overline{G}}(x,y)\leq2$ for every $x,y\in G-S$. Indeed, if $x$ and $y$ are in the same component $C_i$ of $G-S$ for some $i\in[p]$, then pick $z\in C_j$ for some $j\neq i$. Now, $x,y\in N_{\overline{G}}(z)$ which implies $d_{\overline{G}}(x,y)\leq2$. If, otherwise, $x$ and $y$ are in components $C_i$ and $C_j$ of $G-S$ for some distinct $i,j\in[p]$, respectively, then $xy\in E(\overline{G})$ and $d_{\overline{G}}(x,y)=1$. Therefore, either $v=x_i$ and $u\in G-S$, or $v=x_i$ and $u=x_j$ for some $i,j\in[k]$.

Case 1: $S$ is independent. This implies $d_{\overline{G}}(x_i,x_j)=1$ for every $i,j\in[k]$. Thus, $u\in G-S$ and $v=x_i$ for some $i\in[k]$. By symmetry, assume $i=1$. Observe that $G-S$ must contain a nontrivial component. Otherwise, $G$ is a complete bipartite graph and is reconstructible by Theorem <ref>. By symmetry, let $C_1$ be a nontrivial component of $G-S$. We show that $u\in C_1(S)$. First, note that $u$ must be adjacent to every vertex in $S$; otherwise, there exists $x_j$ ($j\neq 1$) in $S$ such that $ux_j\notin E(G)$, which implies $ux_j\in E(\overline{G})$. Since $x_1x_j\in E(\overline{G})$, this means $d_{\overline{G}}(x_1,u)=2$, a contradiction. So, $u\in C_t(S)$ for some $t\in[p]$. Since $G$ is triangle-free and $C_1$ is nontrivial, there exists $x\in C_1$ which is not adjacent to $x_1$. So, if $t\neq1$, then $x_1,u\in N_{\overline{G}}(x)$ and $d_{\overline{G}}(x_1,u)=2$, a contradiction. Thus, $u\in C_1(S)$. By a similar argument, $C_q$ is trivial for each $q\neq 1$. Indeed, if $C_q$ is nontrivial for some $q\neq1$, then there exists $x\in C_q$ which is not adjacent to $x_1$ implying that $d_{\overline{G}}(x_1,u)=2$ (through $x$), a contradiction. So, $C_1$ is the only nontrivial component of $G-S$.

[scale=0.8, every node/.style=scale=0.8]

[thick, rounded corners] (8,1) rectangle (11.2,-0.5) (8,-0.5) rectangle (11.2,-2) (8,-2) rectangle (11.2,-3.5) (8,-3.5) rectangle (11.2,-5) (8,-5) rectangle (11.2,-6.5) (8,-7.8) rectangle (11.2,-9.3);
[thick, decorate, decoration=calligraphic brace, amplitude=3mm] (11.4,1) – (11.4,-10.2);
[thick] (12.2,-4.6) node[scale=1.5] $C_1$;
[thick] (9.6,-6.8) node[fill=black, shape=circle, scale=0.4]  (9.6,-7.1) node[fill=black, shape=circle, scale=0.4]  (9.6,-7.4) node[fill=black, shape=circle, scale=0.4] ;
[thick] (9.6,-9.6) node[fill=black, shape=circle, scale=0.4]  (9.6,-9.9) node[fill=black, shape=circle, scale=0.4]  (9.6,-10.2) node[fill=black, shape=circle, scale=0.4] ;

[thick, rounded corners] (4,-1) rectangle (5.5,-7);

[thick] (C_2) – (x_1) – (C_3) (x_2) – (C_p) – (x_1) (C_2) – (x_2) – (C_3) (C_2) – (x_k) – (C_3) (x_k) – (C_p);
[thick] (8,-1.25) – (x_1) – (8,-2.75) (8,-4.25) – (x_1) – (8,-5.75) (8,-8.55) – (x_1) (8,-4.25) – (x_2) – (8,-1.25) (x_2) – (8,-8.55) (8,-1.25) – (x_k);

[scale=2, fill=white, thick](C_2,C_3,C_p)
[fill=black, thick](dot1,dot2,dot3,dot4,dot5,dot6)
[scale=1.5, left](C_2,C_3,C_p)
[scale=2, fill=white, thick](x_1,x_2,x_k)

[thick] (9.6,0.25) node[scale=1.5] $C_1(\emptyset)$ (9.6,-1.25) node[scale=1.5] $C_1(S)$ (9.6,-2.75) node[scale=1.5] $C_1(1)$ (9.6,-4.25) node[scale=1.5] $C_1(\{1,2\})$ (9.6,-5.75) node[scale=1.5] $C_1(\{1,3\})$ (9.6,-8.55) node[scale=1.5] $C_1(\{1,2,3\})$;

The figure shows the structure of $G$ in Case 1 of Theorem <ref>.

Note that $C_1(\emptyset)\neq\emptyset$. To see this, recall that $C_1(S)\neq\emptyset$ and is an independent set, $C_1$ is a nontrivial component, and $G$ is triangle-free. Therefore, the neighbors in $C_1$ of vertices in $C_1(S)$ can only be in $C_1(\emptyset)$, i.e., $C_1(\emptyset)\neq\emptyset$. Further, every nonempty class in $C_1$ (except for $C_1(\emptyset)$) contains the index 1 (see Figure <ref>); otherwise, as before, there exists a vertex $x$ not adjacent to $x_1$ and $x_1,u\in N_{\overline{G}}(x)$ contradicting $d_{\overline{G}}(x_1,u)=3$. Finally, $u$ must be adjacent to every vertex in $C_1(\emptyset)$; otherwise, there exists $x\in C_1(\emptyset)$ not adjacent to $u$, so $x_1,u\in N_{\overline{G}}(x)$ contradicting $d_{\overline{G}}(x_1,u)=3$. Since $G$ is triangle-free, this implies $C_1(\emptyset)$ is independent. Observe that $G$ is now bipartite as follows: Let every class of $C_1$ except $C_1(\emptyset)$ be in one part along with the trivial components of $G-S$, and let $C_1(\emptyset)$ and $S$ be in the other part. It is easy to check that this forms a bipartition of $G$. Thus, $G$ is reconstructible by Theorem <ref>.

Case 2: $S$ contains an edge. Assume first that $v=x_i$ for some $i\in[k]$ and $u\in G-S$. Let $C_j$ be the component containing $u$ with $j\in[p]$. Since $S$ contains an edge and $G$ is triangle-free, every component of $G-S$ is nontrivial. Pick a component $C_t$ with $t\neq j$. Since $C_t$ is nontrivial, there exists $x\in C_t$ that is not adjacent to $x_i$. Now $x_i,u\in N_{\overline{G}}(x)$ contradicting $d_{\overline{G}}(x_i,u)=3$. Thus, we may assume $v=x_i$ and $u=x_j$ for some $i,j\in[k]$. By symmetry, assume $i=1$ and $j=2$. Since $x_1x_2\in E(G)$ and $G$ is triangle-free, no vertex is adjacent to both $x_1$ and $x_2$. Moreover, every vertex is adjacent to at least one of $x_1$ and $x_2$. Indeed, if some vertex $x$ is nonadjacent to both $x_1$ and $x_2$, then $x_1,x_2\in N_{\overline{G}}(x)$ contradicting $d_{\overline{G}}(x_1,x_2)=3$. Observe that $G$ is again bipartite as follows: Let $N_G(x_1)$ be one part and $N_G(x_2)$ be the other part. Thus, $G$ is reconstructible by Theorem <ref>.

Note that in light of Theorem <ref>, Theorem <ref>  is not needed in order to prove the Reconstruction Conjecture. However, we will refer to this theorem when we consider edge reconstruction in the next section.

Recall that we only need to show that this class is weakly reconstructible. Let $G$ be a a triangle-free graph in $\mathcal{G}_3$ with $\kappa(G)=1$, let $x$ be a cut vertex of $G$, and let $C_1,\dots,C_p$ be the components of $G-x$.

$G-x$ has exactly one nontrivial component, say $C_1$.

If $G-x$ has no nontrivial components, then $G$ is a star contradicting $\diam(G)=~3$. Suppose instead that $C_1$ and $C_2$ are two nontrivial components of $G-x$. Since $G$ is triangle-free, $C_i$ contains at least one vertex, $v_i$, that is not adjacent to $x$ for each $i\in\{1,2\}$. Now $d_G(v_1,v_2)\geq4$, a contradiction. Hence, $G-x$ has exactly one nontrivial component, $C_1$, as desired.

Let $C_1^x(\emptyset):=C_1(\emptyset)$ with respect to cut vertex $x$.$C_1^x$

$C_1$ is bipartite with parts $C_1^x(\emptyset)$ and $C_1(x)$. Furthermore, $G$ is bipartite with parts $C_1^x(\emptyset)\cup \{x\}$ and $C_1(x)\cup T_x$, where $T_x$ denotes the set of all vertices in the trivial components of $G-x$.

Since $G$ is triangle-free, $C_1(x)$ is independent. Also, note that $C_1^x(\emptyset)\neq\emptyset$ because $C_1$ is nontrivial. Since $\diam(\overline{G})=3$, there exist $u,v\in V(G)$ such that $d_{\overline{G}}(u,v)=3$. If $u,v\in C_i$ for some $i\in[p]$, then pick $w\in C_j$ for $j\neq i$. Now $u,v\in N_{\overline{G}}(w)$ and $d_{\overline{G}}(u,v)\leq2$, a contradiction. Similarly, if $u\in C_i$ and $v\in C_j$ for some $i\neq j$, then $uv\in E(\overline{G})$ and $d_{\overline{G}}(u,v)=1$, a contradiction. So, we assume that $u=x$ and $v\in G-x$. Observe that $v\notin C_1^x(\emptyset)$ and $v\notin T_x$; otherwise, $xv\in E(\overline{G})$ and $x,v\in N_{\overline{G}}(w)$ for some $w\in C_1^x(\emptyset)$, respectively. In both cases, $d_{\overline{G}}(x,v)\leq2$, a contradiction. It follows that $v\in C_1(x)$. Furthermore, if there exists $w\in C_1^x(\emptyset)$ such that $vw\notin E(G)$, then $x,v\in N_{\overline{G}}(w)$ and $d_{\overline{G}}(x,v)=2$, a contradiction. Hence, $C_1^x(\emptyset)\subseteq N_G(v)$ which implies $C_1^x(\emptyset)$ is independent since $G$ is triangle-free. Now observe that $C_1$ is bipartite with parts $C_1^x(\emptyset)$ and $C_1(x)$. Further, $G$ is bipartite with parts $C_1^x(\emptyset)\cup \{x\}=N_G(v)$ and $C_1(x)\cup T_x=N_G(x)$, as desired; see Figure <ref>.

[scale=0.8, every node/.style=scale=0.8]

[thick, rounded corners] (6,-2) rectangle (9,-4) (6,-4) rectangle (9,-6);
[thick, decorate, decoration=calligraphic brace, mirror, amplitude=3mm] (9.2,-6) – (9.2,-2);
[thick] (10,-4) node[scale=1.5] $C_1$;

[thick] (C_2) – (x) – (C_3) (C_p) – (x) – (6,-3);

[scale=2, fill=white, thick](C_2,C_3,C_p)
[fill=black, thick](dot1,dot2,dot3)
[scale=1.5, left](C_2,C_3,C_p)
[scale=2, fill=white, thick](x)

[thick] (7.5,-3) node[scale=1.5] $C_1(x)$;
[thick] (7.5,-5) node[scale=1.5] $C_1^x(\emptyset)$;

The figure shows the structure of $G$ in Theorem <ref>.

We now split the rest of the proof into three cases. By slight abuse of notation, we refer to the nontrivial component for any cut vertex as $C_1$.

Case 1: There exists a cut vertex $x$ of $G$ such that $|C_1^x(\emptyset)|\ne |C_1(x)|$. By Claims <ref> and <ref>, this means that there exists a card $H$ of $G$ such that $H$ is disconnected and the only nontrivial component $C_1$ of $H$ is bipartite with unequal parts. Let $H=G-y$ for some cut vertex $y\in V(G)$. By the above claims, $N_G(y)$ consists of all trivial components of $H$, as well as, all vertices in one part of $C_1$. By Lemma <ref>, we can recover $d_G(y)$. Then, since the parts of $C_1$ are unequal, we can identify $N_{C_1}(y)$. Thus, $G$ is reconstructible.

Case 2: $|C_1^{x'}(\emptyset)|=|C_1(x')|$ for every cut vertex $x'$ of $G$ and there exists a cut vertex $x$ of $G$ such that $G-x$ has $k$ trivial components for some integer $k\ge 2$. We claim that $x$ is the only cut vertex of $G$. Indeed, assume some $x'\neq x$ is another cut vertex of $G$ where $G-x'$ has $k'\geq1$ trivial components. Note that $x'\in C_1(x)$ since $G-v$ is connected for every $v\in C_1^x(\emptyset)\cup T_x$. This means $N_G(x')\subseteq C_1^x(\emptyset)\cup \{x\}$, i.e., $d_G(x')\le |C_1^x(\emptyset)|+1$. Further, $d_G(x')=|C_1(x')|+k'$ and $|C_1^{x'}(\emptyset)|=|C_1(x')|=d_G(x')-k'$. Since $|V(G)|=|C_1^{x'}(\emptyset)|+|C_1(x')|+1+k'=|C_1^x(\emptyset)|+|C_1(x)|+1+k$, we have $2(d_G(x')-k')+1+k'=2|C_1^x(\emptyset)|+1+k$ which implies $d_G(x')=|C_1^x(\emptyset)|+(k+k')/2\ge |C_1^x(\emptyset)|+(2+1)/2>|C_1^x(\emptyset)|+1$, a contradiction. Thus, $x$ is the only cut vertex of $G$.

We can identify that $G$ has a unique cut vertex by checking that every card in the deck is connected except for one. Then we pick a card $H=G-z$ for some $z\in V(G)$ with $d_G(z)=1$ (again, $d_G(z)$ is reconstructible by Lemma <ref>). Note that the unique neighbor of any degree 1 vertex in $G$ is a cut vertex, and deleting a degree 1 vertex in $G$ does not create a new cut vertex in the card. So, the unique cut vertex of $G$ is still unique in $G-z$ and is the neighbor of $z$. Thus, $G$ is reconstructible.

Case 3: $|C_1^x(\emptyset)|=|C_1(x)|$ and $G-x$ has exactly one trivial component for every cut vertex $x$ of $G$. Note that $|V(G)|$ is easily reconstructible from the cards of $G$ since each card deletes a single vertex. Since $G-x$ contains a single trivial component, it follows that $|C_1^x(\emptyset)|=|C_1(x)|=(|V(G)|-2)/2$ for every cut vertex $x$ of $G$. By Claim <ref>, $G$ is bipartite with parts $C_1^x(\emptyset)\cup \{x\}$ and $C_1(x)\cup T_x$, where $|T_x|=1$, which implies $|C_1^x(\emptyset)\cup \{x\}|=|C_1(x)\cup T_x|=(|V(G)|-2)/2+1=|V(G)|/2$. Further, $|C_1^x(\emptyset)\cup \{x\}|=|C_1(x)\cup T_x|\geq2$ since $C_1^x(\emptyset)\cup \{x\}$ contains $x$ and at least one non-neighbor of $x$.

Observe that for every cut vertex $x$ of $G$, the trivial component of $G-x$ has degree one in $G$. So, there exists a connected bipartite card $H=G-y$ for some $y\in V(G)$ such that $d_G(y)=1$ (again, $d_G(y)$ is reconstructible by Lemma <ref>). Note that $y$ is the  unique trivial component of $G-z$ for some cut vertex $z$ of $G$; in particular, $z$ is the unique neighbor of $y$ in $G$. Since $H$ is connected, it can be uniquely bipartitioned into parts $X$ and $Y$ with $|X|>|Y|$ and $|X|-|Y|=1$, by the above arguments. Thus, $y$ has no neighbors in $Y$ and the unique neighbor $z$ of $y$ is a vertex in $X$ such that $N_H(z)=Y$. If there exist distinct vertices $x_1$ and $x_2$ in $X$ such that $N_H(x_1)=N_H(x_2)=Y$, then we pick $z$ arbitrarily between $x_1$ and $x_2$ since $H+x_1y$ and $H+x_2y$ are isomorphic. Thus, $G$ is reconstructible.

§ EDGE RECONSTRUCTION: PROOFS OF THEOREMS <REF> AND <REF>

In this section, we consider edge reconstruction for triangle-free graphs in $\mathcal{G}_2\cup\mathcal{G}_3$. The proof of Theorem <ref> uses a systematic approach to try and identify the endpoints of the deleted edge. On the other hand, the proof of Theorem <ref> follows easily from previous results.

Note that this class of graphs is edge-recognizable by Lemmas <ref> and <ref> and Theorem <ref>. So, we only need to show it is weakly edge-reconstructible. Let $G$ be a triangle-free graph in $\mathcal{G}_2$. For every $uv\in E(G)$, if $d_{G-uv}(x,y) \geq 3$ for some $x,y\in V(G-uv)$, then $x=u$ and $y \in N_G[v]$, or $x=v$ and $y \in N_G[u]$; in particular, $d_{G-uv}(u,v) \geq 3$ since $G$ is triangle-free. To see this, note that each pair $x,y\in V(G-uv)$ with $\{x,y\}\neq \{u,v\}$ and $d_{G-uv}(x,y)\geq3$ must be nonadjacent in $G$ and must use the edge $uv$ in $G$ to satisfy $d_G(x,y)=2$.

Pick an edge-card $H=G-uv$ for some $uv\in E(G)$ and let $P_H$$P_H:=\{\{x,y\}: x,y\in V(H)\text{ and }\\ d_H(x,y)\ge3\}$. Assume first that $P_H=\{\{x,y\}\}$, i.e., $|P_H|=1$. Now, $\{x,y\}=\{u,v\}$ and $G=H+xy$ since $\{u,v\}\in P_H$. Assume instead that $|P_H|=k\ge3$. If $\{x,y\}, \{x,z\}\in P_H$, then $x=u$ or $x=v$; otherwise $\{y,z\}=\{u,v\}$ and $x\in N_G(u)\cap N_G(v)$, i.e., $x$, $y$, and $z$ form a triangle, a contradiction. Moreover, since $\{u,v\}\in P_H$, each pair in $P_H$ contains $u$ or $v$, and $k\ge3$, there exist at least two pairs in $P_H$ with a common vertex $x$. So, $x=u$ or $x=v$; say $x=u$. If there exists another vertex $y$ which also appears in more than one pair in $P_H$, then $(x,y)=(u,v)$ and $G=H+xy$. So, suppose $x$ is the only vertex that appears in more than one pair in $P_H$. If $\{y,z\}\in P_H$ with $y\neq x$ and $z\neq x$, then either $y=v$ or $z=v$ (since each pair in $P_H$ contains $u$ or $v$), say $y=v$. Now, $\{x,y\}=\{u,v\}\in P_H$ which means $y$ appears in more than one pair in $P_H$, a contradiction. So, suppose $x$ appears in every pair in $P_H$, i.e., $P_H=\{\{x,y_1\},\{x,y_2\},\dots,\{x,y_k\}\}$. Now, $y_i\in N_G[v]$ for each $i\in[k]$. More precisely, the set $\{y_1,y_2,\dots,y_k,x\}$ forms an induced star in $H$ with center $v=y_i$ for some $i\in[k]$; so, $G=H+xy_i$. Thus, we may assume $|P_H|=2$ for every edge-card $H$.

By symmetry, assume $P_H=\{\{u,v_1\},\{u,v_2\}\}$ where $v\in\{v_1,v_2\}$. Let $w=\{v_1,v_2\}-v$, i.e., $\{v_1,v_2\}=\{v,w\}$. Note that $v_1v_2\in E(H)$ (and therefore, $v_1v_2\in E(G)$). If $d_H(v_1)\neq d_H(v_2)$, then we can identify $v$ since $d_G(v)$ is edge-reconstructible by Lemma <ref>. So, assume that $d_H(v_1)=d_H(v_2)$. Now, there exists an edge-card $H'$ which deletes $v_1v_2$. Interchanging the roles of $u$ and $w$ in the above arguments, $P_{H'}=\{(v,w),(u,w)\}$ since $d_H(u,w)\ge3$ (i.e, $v$ is the only common neighbor of $u$ and $w$ in $G$). So, as in $H$, we may assume $d_{H'}(u)=d_{H'}(v)$. Hence, $d_G(u)=d_G(w)=d_G(v)-1$. This defines a bijection $\sigma:E(G) \rightarrow E(G)$ such that, for every $ab \in E(G)$, there exists $bc \in E(G)$ with $\sigma(ab)=bc$, $\sigma(bc)=ab$, and $d_G(a)=d_G(c)=d_G(b)-1$, where $a,b,c\in V(G)$ and $b$ is the only common neighbor of $a$ and $c$ in $G$. Since every edge in $G$ connects vertices whose degrees differ by one and, therefore, are of different parity, $G$ is bipartite. Thus, $G$ is edge-reconstructible by Theorems <ref> and <ref>.

As before, we only need to show that this class of graphs is weakly edge-reconstructible. Let $G$ be a triangle-free graph in $\mathcal{G}_3$. Since $\diam(G)$ is finite, $G$ is connected. If $\kappa(G)=1$, then we are done by Theorems <ref> and <ref>. If $\kappa(G)=2$, then we are done by Theorems <ref> and <ref>. Finally, if $\kappa(G)\geq3$, then we are done by Theorems <ref> and <ref>. Observe that the result remains true even if $|E(G)|<4$ since the only such graph is $P_4$, and $P_4$ is edge-reconstructible as no other graph can have an edge-card isomorphic to $2K_2$.

§ ACKNOWLEDGMENTS

This project began at the Graduate Research Workshop in Combinatorics in 2021. We sincerely thank the organizers. Alexander Clifton was supported by the Institute for Basic Science (IBS-R029-C1) and partially supported by NSF award DMS-1945200. Xiaonan Liu was partially supported by NSF award DMS-1856645 and NSF award DMS-1954134.

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2014-297 LHCb-PAPER-2014-064 23 December 2014

Determination of the branching fractions of
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}$ and
${{B}^{0}}\\!\rightarrow{{D}^{-}_{s}}{{K}^{+}}$

The LHCb collaboration†††Authors are listed at the end of this paper.

Measurements are presented of the branching fractions of the decays
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}$ and
${{B}^{0}}\\!\rightarrow{{D}^{-}_{s}}{{K}^{+}}$ relative to the decays
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ and
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{\pi}^{+}}$, respectively. The data used
correspond to an integrated luminosity of $3.0\mbox{\,fb}^{-1}$ of proton-
proton collisions. The ratios of branching fractions are

$\dfrac{{\cal B}({{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}})}{{\cal
B}({{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}})}=\,0.0752\pm 0.0015\pm
0.0019$

and

$\dfrac{{\cal B}({{B}^{0}}\\!\rightarrow{{D}^{-}_{s}}{{K}^{+}})}{{\cal
B}({{B}^{0}}\\!\rightarrow{{D}^{-}}{{\pi}^{+}})}=\,0.0129\pm 0.0005\pm
0.0008,$

where the uncertainties are statistical and systematic, respectively.

Submitted to JHEP

© CERN on behalf of the LHCb collaboration, license CC-BY-4.0.

##  1 Introduction

The decays ${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}$ offer a
prime opportunity to measure $C\\!P$ violation in the interference between
mixing and decay [1, 2]. The ${B}^{0}_{s}$ meson can decay into both charge-
conjugate decays, providing sensitivity to the CKM angle $\gamma$ [3]. The
decays ${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ occur predominantly
through colour-allowed tree diagrams (see Fig. 1). A lower bound on the ratio
of the ${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ branching fractions was
derived, ${\cal
B}({{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}})/{\cal
B}({{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}})\geq 0.080\pm 0.007$
[4], with minimal external experimental and theoretical input. Using SU(3)
flavour symmetry, and measurements of
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{\pi}^{+}}$ decays at the $B$-factories, a
prediction for the ratio of branching fractions was calculated, ${\cal
B}({{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}})/{\cal
B}({{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}})=0.086^{+0.009}_{-0.007}$
[4], where the uncertainty includes contributions from non-factorisable
effects [5] and from possible SU(3)-breaking effects of up to 20%.
Contributions from the $W$-exchange diagram, absent in the decay
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$, were estimated from the
${{B}^{0}}\\!\rightarrow{{D}^{-}_{s}}{{K}^{+}}$ branching fraction.

Figure 1:  Feynman diagrams of the processes under study. The upper diagrams
represent the two tree topologies in which a ${B}^{0}_{s}$ meson decays into
the ${{D}^{\mp}_{s}}{{K}^{\pm}}$ final state, and the lower diagrams show the
tree diagram of ${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ and the
$W$-exchange topology of ${B}^{0}_{(s)}\rightarrow{{D}^{-}_{s}}{{K}^{+}}$.

The CDF and Belle collaborations have pioneered the study of this ratio [6,
7], followed by the LHCb collaboration, which measured a ratio lower than the
theoretical bound [8], using data corresponding to an integrated luminosity of
$336\mbox{\,pb}^{-1}$. This paper presents an update for the absolute
branching fraction of ${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}$,
and for the branching fraction of
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}$ relative to that of
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$.

In addition, the ratio of branching fractions of the
${{B}^{0}}\\!\rightarrow{{D}^{-}_{s}}{{K}^{+}}$ and
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{\pi}^{+}}$ decays is reported. The decay
${{B}^{0}}\\!\rightarrow{{D}^{-}_{s}}{{K}^{+}}$ proceeds through the colour-
suppressed $W$-exchange diagram and the branching ratio determination allows
the size of the $W$-exchange amplitude to be estimated, for example in the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}$ decay. The existing
branching fraction measurements by BaBar and Belle, ${\cal
B}({{B}^{0}}\\!\rightarrow{{D}^{-}_{s}}{{K}^{+}})=(2.9\pm
0.4\mathrm{\,(stat)}\pm 0.2\mathrm{\,(syst)})\times 10^{-5}$ [9] and $(1.91\pm
0.24\mathrm{\,(stat)}\pm 0.17\mathrm{\,(syst)})\times 10^{-5}$ [10],
respectively, show a difference of about 1.8 standard deviations, and suggest
that an enhancement of the branching fraction due to rescattering effects is
small [11]. Note that throughout this paper, charge conjugation is implied,
and thus that the branching fraction ${\cal B}(B^{0}_{(s)})$ corresponds to
the average of the branching fraction of the $B^{0}_{(s)}$ decay and the
$\kern 1.79993pt\overline{\kern-1.79993ptB}^{0}_{(s)}$ decay.

The ${p}{p}$-collision data used in this analysis correspond to an integrated
luminosity of $3.0\mbox{\,fb}^{-1}$, of which $1.0\mbox{\,fb}^{-1}$ was
collected by LHCb in 2011 at a centre-of-mass energy of
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, and the remaining
$2.0\mbox{\,fb}^{-1}$ in 2012 at $\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$.

The LHCb detector [12] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream of the
magnet. The polarity of the magnet is reversed periodically throughout data-
taking. The tracking system provides a measurement of momentum, $p$, with a
relative uncertainty that varies from 0.4% at low momentum to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The minimum distance of a track to
a primary vertex, the impact parameter, is measured with a resolution of
$(15+29/\mbox{$p_{\rm T}$}){\,\upmu\rm m}$, where $p_{\rm T}$ is the component
of $p$ transverse to the beam, in ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.
Different types of charged hadrons are distinguished using information from
two ring-imaging detectors.

The trigger consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage, which applies a
full event reconstruction. The software trigger requires a two-, three- or
four-track secondary vertex with a significant displacement from the primary
$pp$ interaction vertices (PVs). At least one charged particle must have a
transverse momentum $\mbox{$p_{\rm
T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and be inconsistent with
originating from the PV. A multivariate algorithm [13] is used for the
identification of secondary vertices consistent with the decay of a $b$
hadron.

In the simulation, $pp$ collisions are generated using Pythia [14,
*Sjostrand:2007gs] with a specific LHCb configuration [16]. Decays of hadronic
particles are described by EvtGen [17]. The interaction of the generated
particles with the detector and its response are implemented using the Geant4
toolkit [18, *Agostinelli:2002hh] as described in Ref. [20].

##  2 Event selection

Candidate ${B}^{0}_{({s})}$ mesons are reconstructed by combining a
${D}^{\pm}_{({s})}$ candidate decaying into three light hadrons,
${{D}^{-}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ or
${{D}^{-}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{-}}$, with an additional
pion or kaon (the “bachelor” particle). Each of the four final-state light
hadrons is required to have a good track quality, high momentum and transverse
momentum, and a large impact parameter with respect to the primary vertex. The
contribution from charmless ${B}^{0}_{({s})}$ decays, such as
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$, is
suppressed by requiring the ${D}^{\pm}_{({s})}$ candidate to have a
significant flight distance from the reconstructed ${B}^{0}_{({s})}$ decay
vertex, and by requiring its mass to fall within a small mass window of
${}^{+22}_{-24}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the
${D}^{\pm}_{({s})}$ mass [21]. To reduce the combinatorial background, a
multivariate algorithm is applied. This boosted decision tree (BDT) [22, 23]
is identical to that used in the analysis of the $C\\!P$ asymmetry in
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}$ decays [3], and was
trained with ${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ candidates
from data, using a weighted data sample based on the sPlot technique [24] as
signal and candidates with an invariant mass greater than
$5445{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ as background. The variables
with the highest discriminating power are found to be the difference between
the $\chi^{2}$ from the vertex fit of the associated PV reconstructed with and
without the considered $b$-hadron candidate, the $p_{\rm T}$ of the final-
state particles, and the angle between the $b$-hadron momentum vector and the
vector connecting its production and decay vertices.

Misidentification of particles leads to peaking backgrounds in the signal
region, for example ${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$
events reconstructed as
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}$ candidates. Pions and
kaons in these decays are required to satisfy particle identification (PID)
requirements, and approximately $60\%$ of the signal is retained while over
$99\%$ of the background is rejected. The efficiencies of these requirements
are determined by studying kinematically selected
${{D}^{*+}}\\!\rightarrow{{D}^{0}}(\\!\rightarrow{{K}^{-}}{{\pi}^{+}}){{\pi}^{+}}$
and ${\mathchar 28931\relax}\\!\rightarrow{p}{{\pi}^{-}}$ decays obtained from
data, which provide high-purity PID.

Table 1: Kinematic and PID selection efficiencies for each signal decay, as determined from simulated events and data, respectively. The kinematic efficiencies represent weighted averages determined from events simulated at $\sqrt{s}=7\mathrm{\,Te\kern-0.90005ptV}$ ($34\%$) and $\sqrt{s}=8\mathrm{\,Te\kern-0.90005ptV}$ ($66\%$). The binomial uncertainties result from the size of the simulated samples. Selection efficiency (%) | Kinematic | PID | Total
---|---|---|---
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{\pi}^{+}}$ | $1.89\pm 0.01$ | $74.29\pm 0.07$ | $1.40\pm 0.01$
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ | $1.92\pm 0.02$ | $67.10\pm 0.09$ | $1.29\pm 0.01$
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}$ | $2.08\pm 0.01$ | $55.52\pm 0.17$ | $1.15\pm 0.01$
${{B}^{0}}\\!\rightarrow{{D}^{-}_{s}}{{K}^{+}}$ | $1.70\pm 0.03$ | $58.11\pm 0.82$ | $0.99\pm 0.02$

In the ${{B}^{0}}\\!\rightarrow{{D}^{-}}{{\pi}^{+}}$ selection, loose PID
requirements are applied since the branching fraction of the signal process is
much larger than those of background decays resulting from misidentification.
In the ${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}$ selections, a stricter
requirement is applied to the ${D}^{+}_{s}$ decays to distinguish
${D}^{+}_{s}$ and ${D}^{+}$ mesons. For these decays, a further selection
requirement is applied to reduce the background from ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{-}}$ decays, where one of the ${D}^{+}_{s}$
daughters is misidentified as a proton. This requirement removes any candidate
which fulfils two criteria: that there is a large probability for one of the
${D}^{+}_{s}$ daughters to be a misidentified proton, and that when the
${D}^{+}_{s}$ decay is reconstructed under the ${\mathchar
28931\relax}^{+}_{c}$ hypothesis, its invariant mass falls within
$21{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the nominal ${\mathchar
28931\relax}^{+}_{c}$ mass [21]. This procedure almost fully eliminates this
background. The efficiency of the selection is obtained from simulation and is
summarised in Table 1.

##  3 Signal yield determination

An unbinned maximum likelihood fit to the candidate invariant mass
distribution is performed for each of the three final states. The signal
shapes are parametrised by a double-sided Crystal Ball shape [25]. This
function consists of a central Gaussian part, whose mean and width are free
parameters, and power-law tails on both lower and upper sides, to account for
energy loss due to final-state radiation and detector resolution effects. The
functional form for the combinatorial background, an exponential function with
an offset, is obtained from same-charge ${{D}^{\pm}_{s}}{{\pi}^{\pm}}$
combinations. All parameters of the combinatorial background are left free in
the fit to data.

The physical backgrounds can be split into two categories: misidentified
backgrounds, predominantly where one of the final state pions (kaons) is
mistaken for a kaon (pion); and partially reconstructed backgrounds, where a
neutral pion or a photon is not included in the candidate reconstruction,
causing the reconstructed ${B}^{0}_{({s})}$ mass to shift to lower values.
Some backgrounds fall into both categories. The number of background
components considered varies per final state. The invariant mass shapes of
these backgrounds are obtained from simulation at
$\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$, with the event selection applied.
The yield of each background is a parameter in the fit, with most background
components Gaussian-constrained around the expected yield normalised to the
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{\pi}^{+}}$ yield obtained from data. The
constraints are assigned an uncertainty of $10\%$, which reflects the
uncertainties from production fractions, branching fractions, and
reconstruction efficiencies. The resulting background yields from the fit are
close to the expected values. The results of the fits for the three final
states are shown in Figs. 2 and 3, and in Table 2. The three fits are
independent, and no parameters are shared among them.

(a)
(b)

Figure 2:  Results of the fits to the invariant mass distributions of the final states (a) ${{D}^{-}}{{\pi}^{+}}$ and (b) ${{D}^{-}_{s}}{{\pi}^{+}}$. Figure 3:  Result of the fit to the invariant mass distribution of the final state ${{D}^{\mp}_{s}}{{K}^{\pm}}$. Table 2: Yields for the four signal decay types, as obtained from the fits. | Yield
---|---
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{\pi}^{+}}$ | $458\,940\pm 959$
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ | $75\,566\pm 342$
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}$ | $5\,101\pm 100$
${{B}^{0}}\\!\rightarrow{{D}^{-}_{s}}{{K}^{+}}$ | $2\,452\pm\hphantom{0}98$

Various consistency checks are performed for each of the fits. The fitted
yield of ${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ events
reconstructed in the ${D}^{\mp}_{s}$ ${K}^{\pm}$ final state, which is allowed
to vary in the fit, is consistent with the expected yield based on the
relative branching fraction, particle misidentification probability and
reconstruction efficiency. For each of the fits, consistency is also found
between the fitted yield for both magnet polarities separately and the
fraction of data corresponding to that polarity. This demonstrates that the
relative yields are stable as a function of time and magnet polarity.

##  4 Systematic uncertainties

Systematic uncertainties arise from the fit model and the candidate selection,
and are summarised in Table 3. The systematic uncertainty from the fit model
is determined by applying variations to the fit model and comparing the yield
to the nominal result, taking the difference as a systematic uncertainty.
These variations include a different combinatorial shape, fixing the signal
shape tail parameters to values obtained from simulation, and using background
shapes determined from simulation matching the LHCb conditions during 2011
($\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$). In the ${{D}^{-}}{{\pi}^{+}}$
analysis, the fit range is reduced to start at
$5100{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. In the
${{D}^{\mp}_{s}}{{K}^{\pm}}$ analysis, the as yet unobserved decay
${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{{D}^{-}_{s}}{p}$ is omitted
from the fit.

The uncertainty on the candidate selection is separated into three parts: the
uncertainty due to the differences between data and simulation, that due to
the PID requirements on the final state pions and kaons, and that due to the
hardware trigger efficiency. The first of these uncertainties is determined
from the selection efficiency difference between magnet polarities in
simulation, and by estimating the uncertainty on the BDT selection efficiency
due to differences between data and simulation. This is calculated by
reweighting simulated events to match the data more closely, and calculating
the difference in BDT efficiency between those and the unweighted samples. The
uncertainty on the PID efficiency and misidentification rate is estimated by
comparing the PID performance measured using a simulated ${D}^{*}$ calibration
sample with that observed in simulated signal events. The systematic
uncertainty from the hardware trigger efficiency arises from differences in
the pion and kaon trigger efficiencies which are not reproduced in the
simulation. The uncertainty is scaled with the fraction of events where a
signal track was responsible for triggering.

A further systematic uncertainty is added to account for possible charmless
${B}^{0}$ decays peaking under the
${{B}^{0}}\\!\rightarrow{{D}^{-}_{s}}{{K}^{+}}$ signal. Some of the
uncertainties cancel in the ratios of branching fractions, leading to lower
overall systematic uncertainties than those determined individually for each
decay channel.

Table 3: Systematic uncertainties on the ratios of branching fractions, in $\%$, obtained as described in the text. The total uncertainty is obtained by adding the separate contributions in quadrature. Source | $\dfrac{{{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}}{{{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}}$ | $\dfrac{{{B}^{0}}\\!\rightarrow{{D}^{-}_{s}}{{K}^{+}}}{{{B}^{0}}\\!\rightarrow{{D}^{-}}{{\pi}^{+}}}$
---|---|---
Fit model | $1.1$ | $3.6$
Candidate selection | $2.1$ | $2.9$
Hardware trigger | $1.0$ | $1.2$
Charmless background | – | $1.0$
Total | $2.5$ | $4.9$

##  5 Determination of branching fractions

The ratios of branching fractions are evaluated using the expression

$\dfrac{{\cal B}(A)}{{\cal
B}(B)}=\dfrac{\varepsilon_{B}}{\varepsilon_{A}}\dfrac{N_{A}}{N_{B}}\dfrac{f_{B}}{f_{A}}\dfrac{{\cal
B}_{{D}^{\pm}_{({s})}}}{{\cal B}_{{D}^{\pm}_{({s})}}},$ (1)

where $\varepsilon_{X}$, $f_{X}$ and $N_{X}$ are the selection efficiency, the
hadronisation fraction, and the fitted yield of decay $X$, respectively, and
${\cal B}_{{D}^{\pm}_{({s})}}$ is the branching fraction of
${D}^{\pm}_{({s})}$ decays, as appropriate. The following values are used as
input [21]:

$\displaystyle{\cal B}({{B}^{0}}\\!\rightarrow{{D}^{-}}{{\pi}^{+}})=$
$\displaystyle\,(2.68\pm 0.13)\times 10^{-3},$ $\displaystyle{\cal
B}({{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}})=$
$\displaystyle\,(3.04\pm 0.23)\times 10^{-3},$ $\displaystyle{\cal
B}({{D}^{-}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}})=$
$\displaystyle\,(9.13\pm 0.19)\times 10^{-2},$ $\displaystyle{\cal
B}({{D}^{-}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{-}})=$
$\displaystyle\,(5.39\pm 0.21)\times 10^{-2}.$

As a cross-check, a value ${\cal
B}({{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}})=(2.95\pm
0.01\mathrm{\,(stat)})\times 10^{-3}$ was obtained from the measured
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ and
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{\pi}^{+}}$ yields using Eq. (1). This
measurement is compatible with the world-average value, and the central value
is unchanged with respect to the previous result published by LHCb [8].

The following results are obtained

$\displaystyle\dfrac{{\cal
B}({{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}})}{{\cal
B}({{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}})}=$
$\displaystyle\,0.0752\pm 0.0015\mathrm{\,(stat)}\pm 0.0019\mathrm{\,(syst)},$
$\displaystyle{\cal
B}({{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}})=$
$\displaystyle\,(2.29\pm 0.05\mathrm{\,(stat)}\pm 0.06\mathrm{\,(syst)}\pm
0.17({\cal B}_{{{B}^{0}_{s}}}))\times 10^{-4},$ $\displaystyle\dfrac{{\cal
B}({{B}^{0}}\\!\rightarrow{{D}^{-}_{s}}{{K}^{+}})}{{\cal
B}({{B}^{0}}\\!\rightarrow{{D}^{-}}{{\pi}^{+}})}=$ $\displaystyle\,0.0129\pm
0.0005\mathrm{\,(stat)}\pm 0.0007\mathrm{\,(syst)}\pm 0.0004({\cal
B}_{{D}^{\pm}_{({s})}}),$ $\displaystyle{\cal
B}({{B}^{0}}\\!\rightarrow{{D}^{-}_{s}}{{K}^{+}})=$ $\displaystyle\,(3.45\pm
0.14\mathrm{\,(stat)}\pm 0.20\mathrm{\,(syst)}\pm 0.20({\cal
B}_{{{B}^{0}},{D}^{\pm}_{({s})}}))\times 10^{-5},$

where the uncertainties labelled $({\cal B})$ arise from the uncertainties on
the branching fractions used as input.

The branching fractions of
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}$ and
${{B}^{0}}\\!\rightarrow{{D}^{-}_{s}}{{K}^{+}}$ presented here are more
precise than the current world-average values. The result for ${\cal
B}({{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}})/{\cal
B}({{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}})$ is compatible with
theoretical expectations [4]. As expected [5], the branching fraction of the
decay ${{B}^{0}}\\!\rightarrow{{D}^{-}_{s}}{{K}^{+}}$, dominated by the
$W$-exchange topology, is suppressed compared to the decay
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{\pi}^{+}}$, which predominantly proceeds
through the colour-allowed tree topology. The measured value of ${\cal
B}({{B}^{0}}\\!\rightarrow{{D}^{-}_{s}}{{K}^{+}})$ is in good agreement with
existing measurements from the BaBar collaboration [9], and is larger than the
result published by the Belle collaboration [10] with a significance of more
than three standard deviations.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); INFN (Italy);
FOM and NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania);
MinES and FANO (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU
(Ukraine); STFC (United Kingdom); NSF (USA). The Tier1 computing centres are
supported by IN2P3 (France), KIT and BMBF (Germany), INFN (Italy), NWO and
SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom). We are indebted
to the communities behind the multiple open source software packages on which
we depend. We are also thankful for the computing resources and the access to
software R&D tools provided by Yandex LLC (Russia). Individual groups or
members have received support from EPLANET, Marie Skłodowska-Curie Actions and
ERC (European Union), Conseil général de Haute-Savoie, Labex ENIGMASS and
OCEVU, Région Auvergne (France), RFBR (Russia), XuntaGal and GENCAT (Spain),
Royal Society and Royal Commission for the Exhibition of 1851 (United
Kingdom).

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LHCb collaboration

R. Aaij41, B. Adeva37, M. Adinolfi46, A. Affolder52, Z. Ajaltouni5, S. Akar6,
J. Albrecht9, F. Alessio38, M. Alexander51, S. Ali41, G. Alkhazov30, P.
Alvarez Cartelle37, A.A. Alves Jr25,38, S. Amato2, S. Amerio22, Y. Amhis7, L.
An3, L. Anderlini17,g, J. Anderson40, R. Andreassen57, M. Andreotti16,f, J.E.
Andrews58, R.B. Appleby54, O. Aquines Gutierrez10, F. Archilli38, A.
Artamonov35, M. Artuso59, E. Aslanides6, G. Auriemma25,n, M. Baalouch5, S.
Bachmann11, J.J. Back48, A. Badalov36, C. Baesso60, W. Baldini16, R.J.
Barlow54, C. Barschel38, S. Barsuk7, W. Barter47, V. Batozskaya28, V.
Battista39, A. Bay39, L. Beaucourt4, J. Beddow51, F. Bedeschi23, I. Bediaga1,
L.J. Bel41, S. Belogurov31, K. Belous35, I. Belyaev31, E. Ben-Haim8, G.
Bencivenni18, S. Benson38, J. Benton46, A. Berezhnoy32, R. Bernet40, A.
Bertolin22, M.-O. Bettler47, M. van Beuzekom41, A. Bien11, S. Bifani45, T.
Bird54, A. Bizzeti17,i, P.M. Bjørnstad54, T. Blake48, F. Blanc39, J. Blouw10,
S. Blusk59, V. Bocci25, A. Bondar34, N. Bondar30,38, W. Bonivento15, S.
Borghi54, A. Borgia59, M. Borsato7, T.J.V. Bowcock52, E. Bowen40, C. Bozzi16,
D. Brett54, M. Britsch10, T. Britton59, J. Brodzicka54, N.H. Brook46, A.
Bursche40, J. Buytaert38, S. Cadeddu15, R. Calabrese16,f, M. Calvi20,k, M.
Calvo Gomez36,p, P. Campana18, D. Campora Perez38, L. Capriotti54, A.
Carbone14,d, G. Carboni24,l, R. Cardinale19,38,j, A. Cardini15, L. Carson50,
K. Carvalho Akiba2,38, RCM Casanova Mohr36, G. Casse52, L. Cassina20,k, L.
Castillo Garcia38, M. Cattaneo38, Ch. Cauet9, R. Cenci23,t, M. Charles8, Ph.
Charpentier38, M. Chefdeville4, S. Chen54, S.-F. Cheung55, N. Chiapolini40, M.
Chrzaszcz40,26, X. Cid Vidal38, G. Ciezarek41, P.E.L. Clarke50, M.
Clemencic38, H.V. Cliff47, J. Closier38, V. Coco38, J. Cogan6, E. Cogneras5,
V. Cogoni15, L. Cojocariu29, G. Collazuol22, P. Collins38, A. Comerma-
Montells11, A. Contu15,38, A. Cook46, M. Coombes46, S. Coquereau8, G. Corti38,
M. Corvo16,f, I. Counts56, B. Couturier38, G.A. Cowan50, D.C. Craik48, A.C.
Crocombe48, M. Cruz Torres60, S. Cunliffe53, R. Currie53, C. D’Ambrosio38, J.
Dalseno46, P. David8, P.N.Y. David41, A. Davis57, K. De Bruyn41, S. De
Capua54, M. De Cian11, J.M. De Miranda1, L. De Paula2, W. De Silva57, P. De
Simone18, C.-T. Dean51, D. Decamp4, M. Deckenhoff9, L. Del Buono8, N.
Déléage4, D. Derkach55, O. Deschamps5, F. Dettori38, B. Dey40, A. Di Canto38,
A Di Domenico25, F. Di Ruscio24, H. Dijkstra38, S. Donleavy52, F. Dordei11, M.
Dorigo39, A. Dosil Suárez37, D. Dossett48, A. Dovbnya43, K. Dreimanis52, G.
Dujany54, F. Dupertuis39, P. Durante38, R. Dzhelyadin35, A. Dziurda26, A.
Dzyuba30, S. Easo49,38, U. Egede53, V. Egorychev31, S. Eidelman34, S.
Eisenhardt50, U. Eitschberger9, R. Ekelhof9, L. Eklund51, I. El Rifai5, Ch.
Elsasser40, S. Ely59, S. Esen11, H.-M. Evans47, T. Evans55, A. Falabella14, C.
Färber11, C. Farinelli41, N. Farley45, S. Farry52, R. Fay52, D. Ferguson50, V.
Fernandez Albor37, F. Ferreira Rodrigues1, M. Ferro-Luzzi38, S. Filippov33, M.
Fiore16,f, M. Fiorini16,f, M. Firlej27, C. Fitzpatrick39, T. Fiutowski27, P.
Fol53, M. Fontana10, F. Fontanelli19,j, R. Forty38, O. Francisco2, M. Frank38,
C. Frei38, M. Frosini17, J. Fu21,38, E. Furfaro24,l, A. Gallas Torreira37, D.
Galli14,d, S. Gallorini22,38, S. Gambetta19,j, M. Gandelman2, P. Gandini59, Y.
Gao3, J. García Pardiñas37, J. Garofoli59, J. Garra Tico47, L. Garrido36, D.
Gascon36, C. Gaspar38, U. Gastaldi16, R. Gauld55, L. Gavardi9, G. Gazzoni5, A.
Geraci21,v, E. Gersabeck11, M. Gersabeck54, T. Gershon48, Ph. Ghez4, A.
Gianelle22, S. Gianì39, V. Gibson47, L. Giubega29, V.V. Gligorov38, C.
Göbel60, D. Golubkov31, A. Golutvin53,31,38, A. Gomes1,a, C. Gotti20,k, M.
Grabalosa Gándara5, R. Graciani Diaz36, L.A. Granado Cardoso38, E. Graugés36,
E. Graverini40, G. Graziani17, A. Grecu29, E. Greening55, S. Gregson47, P.
Griffith45, L. Grillo11, O. Grünberg63, B. Gui59, E. Gushchin33, Yu. Guz35,38,
T. Gys38, C. Hadjivasiliou59, G. Haefeli39, C. Haen38, S.C. Haines47, S.
Hall53, B. Hamilton58, T. Hampson46, X. Han11, S. Hansmann-Menzemer11, N.
Harnew55, S.T. Harnew46, J. Harrison54, J. He38, T. Head39, V. Heijne41, K.
Hennessy52, P. Henrard5, L. Henry8, J.A. Hernando Morata37, E. van
Herwijnen38, M. Heß63, A. Hicheur2, D. Hill55, M. Hoballah5, C. Hombach54, W.
Hulsbergen41, N. Hussain55, D. Hutchcroft52, D. Hynds51, M. Idzik27, P.
Ilten56, R. Jacobsson38, A. Jaeger11, J. Jalocha55, E. Jans41, A. Jawahery58,
F. Jing3, M. John55, D. Johnson38, C.R. Jones47, C. Joram38, B. Jost38, N.
Jurik59, S. Kandybei43, W. Kanso6, M. Karacson38, T.M. Karbach38, S.
Karodia51, M. Kelsey59, I.R. Kenyon45, T. Ketel42, B. Khanji20,38,k, C.
Khurewathanakul39, S. Klaver54, K. Klimaszewski28, O. Kochebina7, M. Kolpin11,
I. Komarov39, R.F. Koopman42, P. Koppenburg41,38, M. Korolev32, L. Kravchuk33,
K. Kreplin11, M. Kreps48, G. Krocker11, P. Krokovny34, F. Kruse9, W.
Kucewicz26,o, M. Kucharczyk20,26,k, V. Kudryavtsev34, K. Kurek28, T.
Kvaratskheliya31, V.N. La Thi39, D. Lacarrere38, G. Lafferty54, A. Lai15, D.
Lambert50, R.W. Lambert42, G. Lanfranchi18, C. Langenbruch48, B. Langhans38,
T. Latham48, C. Lazzeroni45, R. Le Gac6, J. van Leerdam41, J.-P. Lees4, R.
Lefèvre5, A. Leflat32, J. Lefrançois7, O. Leroy6, T. Lesiak26, B.
Leverington11, Y. Li7, T. Likhomanenko64, M. Liles52, R. Lindner38, C. Linn38,
F. Lionetto40, B. Liu15, S. Lohn38, I. Longstaff51, J.H. Lopes2, P. Lowdon40,
D. Lucchesi22,r, H. Luo50, A. Lupato22, E. Luppi16,f, O. Lupton55, F.
Machefert7, I.V. Machikhiliyan31, F. Maciuc29, O. Maev30, S. Malde55, A.
Malinin64, G. Manca15,e, G. Mancinelli6, A. Mapelli38, J. Maratas5, J.F.
Marchand4, U. Marconi14, C. Marin Benito36, P. Marino23,t, R. Märki39, J.
Marks11, G. Martellotti25, M. Martinelli39, D. Martinez Santos42, F. Martinez
Vidal65, D. Martins Tostes2, A. Massafferri1, R. Matev38, Z. Mathe38, C.
Matteuzzi20, A. Mazurov45, M. McCann53, J. McCarthy45, A. McNab54, R.
McNulty12, B. McSkelly52, B. Meadows57, F. Meier9, M. Meissner11, M. Merk41,
D.A. Milanes62, M.-N. Minard4, N. Moggi14, J. Molina Rodriguez60, S. Monteil5,
M. Morandin22, P. Morawski27, A. Mordà6, M.J. Morello23,t, J. Moron27, A.-B.
Morris50, R. Mountain59, F. Muheim50, K. Müller40, M. Mussini14, B. Muster39,
P. Naik46, T. Nakada39, R. Nandakumar49, I. Nasteva2, M. Needham50, N. Neri21,
S. Neubert38, N. Neufeld38, M. Neuner11, A.D. Nguyen39, T.D. Nguyen39, C.
Nguyen-Mau39,q, M. Nicol7, V. Niess5, R. Niet9, N. Nikitin32, T. Nikodem11, A.
Novoselov35, D.P. O’Hanlon48, A. Oblakowska-Mucha27, V. Obraztsov35, S.
Ogilvy51, O. Okhrimenko44, R. Oldeman15,e, C.J.G. Onderwater66, M. Orlandea29,
B. Osorio Rodrigues1, J.M. Otalora Goicochea2, A. Otto38, P. Owen53, A.
Oyanguren65, B.K. Pal59, A. Palano13,c, F. Palombo21,u, M. Palutan18, J.
Panman38, A. Papanestis49,38, M. Pappagallo51, L.L. Pappalardo16,f, C.
Parkes54, C.J. Parkinson9,45, G. Passaleva17, G.D. Patel52, M. Patel53, C.
Patrignani19,j, A. Pearce54,49, A. Pellegrino41, G. Penso25,m, M. Pepe
Altarelli38, S. Perazzini14,d, P. Perret5, L. Pescatore45, E. Pesen67, K.
Petridis53, A. Petrolini19,j, E. Picatoste Olloqui36, B. Pietrzyk4, T.
Pilař48, D. Pinci25, A. Pistone19, S. Playfer50, M. Plo Casasus37, F. Polci8,
A. Poluektov48,34, I. Polyakov31, E. Polycarpo2, A. Popov35, D. Popov10, B.
Popovici29, C. Potterat2, E. Price46, J.D. Price52, J. Prisciandaro39, A.
Pritchard52, C. Prouve46, V. Pugatch44, A. Puig Navarro39, G. Punzi23,s, W.
Qian4, R Quagliani7,46, B. Rachwal26, J.H. Rademacker46, B.
Rakotomiaramanana39, M. Rama23, M.S. Rangel2, I. Raniuk43, N. Rauschmayr38, G.
Raven42, F. Redi53, S. Reichert54, M.M. Reid48, A.C. dos Reis1, S.
Ricciardi49, S. Richards46, M. Rihl38, K. Rinnert52, V. Rives Molina36, P.
Robbe7, A.B. Rodrigues1, E. Rodrigues54, P. Rodriguez Perez54, S. Roiser38, V.
Romanovsky35, A. Romero Vidal37, M. Rotondo22, J. Rouvinet39, T. Ruf38, H.
Ruiz36, P. Ruiz Valls65, J.J. Saborido Silva37, N. Sagidova30, P. Sail51, B.
Saitta15,e, V. Salustino Guimaraes2, C. Sanchez Mayordomo65, B. Sanmartin
Sedes37, R. Santacesaria25, C. Santamarina Rios37, E. Santovetti24,l, A.
Sarti18,m, C. Satriano25,n, A. Satta24, D.M. Saunders46, D. Savrina31,32, M.
Schiller38, H. Schindler38, M. Schlupp9, M. Schmelling10, B. Schmidt38, O.
Schneider39, A. Schopper38, M.-H. Schune7, R. Schwemmer38, B. Sciascia18, A.
Sciubba25,m, A. Semennikov31, I. Sepp53, N. Serra40, J. Serrano6, L.
Sestini22, P. Seyfert11, M. Shapkin35, I. Shapoval16,43,f, Y. Shcheglov30, T.
Shears52, L. Shekhtman34, V. Shevchenko64, A. Shires9, R. Silva Coutinho48, G.
Simi22, M. Sirendi47, N. Skidmore46, I. Skillicorn51, T. Skwarnicki59, N.A.
Smith52, E. Smith55,49, E. Smith53, J. Smith47, M. Smith54, H. Snoek41, M.D.
Sokoloff57, F.J.P. Soler51, F. Soomro39, D. Souza46, B. Souza De Paula2, B.
Spaan9, P. Spradlin51, S. Sridharan38, F. Stagni38, M. Stahl11, S. Stahl11, O.
Steinkamp40, O. Stenyakin35, F Sterpka59, S. Stevenson55, S. Stoica29, S.
Stone59, B. Storaci40, S. Stracka23,t, M. Straticiuc29, U. Straumann40, R.
Stroili22, L. Sun57, W. Sutcliffe53, K. Swientek27, S. Swientek9, V.
Syropoulos42, M. Szczekowski28, P. Szczypka39,38, T. Szumlak27, S. T’Jampens4,
M. Teklishyn7, G. Tellarini16,f, F. Teubert38, C. Thomas55, E. Thomas38, J.
van Tilburg41, V. Tisserand4, M. Tobin39, J. Todd57, S. Tolk42, L.
Tomassetti16,f, D. Tonelli38, S. Topp-Joergensen55, N. Torr55, E. Tournefier4,
S. Tourneur39, M.T. Tran39, M. Tresch40, A. Trisovic38, A. Tsaregorodtsev6, P.
Tsopelas41, N. Tuning41, M. Ubeda Garcia38, A. Ukleja28, A. Ustyuzhanin64, U.
Uwer11, C. Vacca15, V. Vagnoni14, G. Valenti14, A. Vallier7, R. Vazquez
Gomez18, P. Vazquez Regueiro37, C. Vázquez Sierra37, S. Vecchi16, J.J.
Velthuis46, M. Veltri17,h, G. Veneziano39, M. Vesterinen11, JVVB Viana
Barbosa38, B. Viaud7, D. Vieira2, M. Vieites Diaz37, X. Vilasis-Cardona36,p,
A. Vollhardt40, D. Volyanskyy10, D. Voong46, A. Vorobyev30, V. Vorobyev34, C.
Voß63, J.A. de Vries41, R. Waldi63, C. Wallace48, R. Wallace12, J. Walsh23, S.
Wandernoth11, J. Wang59, D.R. Ward47, N.K. Watson45, D. Websdale53, M.
Whitehead48, D. Wiedner11, G. Wilkinson55,38, M. Wilkinson59, M.P. Williams45,
M. Williams56, H.W. Wilschut66, F.F. Wilson49, J. Wimberley58, J. Wishahi9, W.
Wislicki28, M. Witek26, G. Wormser7, S.A. Wotton47, S. Wright47, K. Wyllie38,
Y. Xie61, Z. Xing59, Z. Xu39, Z. Yang3, X. Yuan3, O. Yushchenko35, M.
Zangoli14, M. Zavertyaev10,b, L. Zhang3, W.C. Zhang12, Y. Zhang3, A.
Zhelezov11, A. Zhokhov31, L. Zhong3.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Milano, Milano, Italy
22Sezione INFN di Padova, Padova, Italy
23Sezione INFN di Pisa, Pisa, Italy
24Sezione INFN di Roma Tor Vergata, Roma, Italy
25Sezione INFN di Roma La Sapienza, Roma, Italy
26Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
27AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
28National Center for Nuclear Research (NCBJ), Warsaw, Poland
29Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
30Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
31Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
32Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
33Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
34Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
35Institute for High Energy Physics (IHEP), Protvino, Russia
36Universitat de Barcelona, Barcelona, Spain
37Universidad de Santiago de Compostela, Santiago de Compostela, Spain
38European Organization for Nuclear Research (CERN), Geneva, Switzerland
39Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
40Physik-Institut, Universität Zürich, Zürich, Switzerland
41Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
42Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
43NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
44Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
45University of Birmingham, Birmingham, United Kingdom
46H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
47Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
48Department of Physics, University of Warwick, Coventry, United Kingdom
49STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
50School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
51School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
52Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
53Imperial College London, London, United Kingdom
54School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
55Department of Physics, University of Oxford, Oxford, United Kingdom
56Massachusetts Institute of Technology, Cambridge, MA, United States
57University of Cincinnati, Cincinnati, OH, United States
58University of Maryland, College Park, MD, United States
59Syracuse University, Syracuse, NY, United States
60Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
61Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
62Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
63Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
64National Research Centre Kurchatov Institute, Moscow, Russia, associated to
31
65Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 36
66Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 41
67Celal Bayar University, Manisa, Turkey, associated to 38

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, Italy
hUniversità di Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità di Roma La Sapienza, Roma, Italy
nUniversità della Basilicata, Potenza, Italy
oAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
pLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
qHanoi University of Science, Hanoi, Viet Nam
rUniversità di Padova, Padova, Italy
sUniversità di Pisa, Pisa, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità degli Studi di Milano, Milano, Italy
vPolitecnico di Milano, Milano, Italy