# Improved Duplication-Transfer-Loss Reconciliation with
## 2 Definitions and Preliminaries
### DTLx Reconciliation
(**d**) An optimal DTL reconciliation of \(G\) and \(S\) illustrating how the gene tree (blue lines) may have evolved within the species tree (tubes) without the use of \(\mathbb{TL}\) or \(\mathbb{TX}\) events. This DTL reconciliation invokes two transfer events and two losses. All reconciliations shown in this figure use event costs of 1, 2, 3, 4, and 3 for losses, duplications, transfers, \(\mathbb{TL}\), and \(\mathbb{TX}\) events, respectively. Thus, we get a total reconciliation cost of 8 for this optimal DTL reconciliation. (**e**) An optimal reconciliation of \(G\) and \(S\) that uses \(\mathbb{TL}\) events but not \(\mathbb{TX}\) events. This reconciliation utilizes the hidden node h in \(G^{\prime}\) to facilitate a \(\mathbb{TL}\) event from species E to species B. It invokes one transfer event and one \(\mathbb{TL}\) event, for a total reconciliation cost of 7. (**f**) An optimal DTLx reconciliation of \(G\) and \(S\). This DTLx reconciliation utilizes the hidden node h in \(G^{\prime}\) to facilitate a \(\mathbb{TX}\) event. It invokes one transfer event and one \(\mathbb{TX}\) event, for a total reconciliation cost of 6. Accordingly, the reconciliation shown in (**f**) makes use of an extra leaf and extra node on \(S^{\prime}\) (dotted green tube). **Definition 6** (Losses). : _Given a DTLx reconciliation \(\alpha=\langle\mathcal{L},G^{\prime},S^{\prime},\mathcal{M},\Sigma,\Delta,\Theta, \Theta_{L},\Theta_{X},\Xi,\tau\rangle\) for \(G\) and \(S\), let \(g\in V(G^{\prime})\) and \(\{g^{\prime},g^{\prime\prime}\}=Ch(g)\) if \(g\in I(G)\cap I(G^{\prime})\), and \(\{g^{\prime}\}=Ch(g)\) otherwise. The minimum number of losses \(\mathrm{Loss}_{\mathrm{a}}(g)\) at node \(g\) (or, more accurately, the minimum number of losses incurred along the child edge(s) of \(g\)) is defined to be:_  
* \(d_{S^{\prime}}(\mathcal{M}(g),\mathcal{M}(g^{\prime}))+d_{S^{\prime}}( \mathcal{M}(g),\)__\((\mathcal{M}(g^{\prime\prime}))-2,\) _if_ \(g\in\Sigma\)_;_
* \(d_{S^{\prime}}(\mathcal{M}(g),\mathcal{M}(g^{\prime}))+d_{S^{\prime}}( \mathcal{M}(g),\)__\(\mathcal{M}(g^{\prime\prime}))\) _if_ \(g\in\Delta\)_;_
* \(d_{S^{\prime}}(\mathcal{M}(g),\mathcal{M}(g^{\prime\prime}))+d_{S^{\prime}}( \tau(g),\mathcal{M}(g^{\prime}))\) _if_ \((g,g^{\prime})\in\Xi\) _and_ \(g\in\Theta,\) _and;_
* \(d_{S^{\prime}}(\tau(g),\mathcal{M}(g^{\prime})\) _if_ \((g,g^{\prime})\in\Xi\)_, and_ \(g\in\Theta_{L}\) _or_ \(g\in\Theta_{X}\)_._  
_The total number of losses associated with reconciliation \(\alpha\), denoted \(\mathrm{Loss}_{\alpha}\), is defined to be \(\sum_{g\in I(G^{\prime})}\mathrm{Loss}_{\mathrm{a}}(g)\)._  
Note that the implicit gene loss (of the donor's copy of the gene) associated with \(\mathbb{TL}\) or \(\mathbb{TX}\) events is not counted in the definition of losses above. Instead, \(\mathbb{TL}\) and \(\mathbb{TX}\) events can be assigned a higher event cost than a standard transfer event (see next paragraph) to account for these implicit losses. In the DTLx reconciliation framework, each event other than speciation (which is considered a null event) is assigned a positive cost. Let \(P_{\Delta}\), \(P_{\Theta}\), \(P_{\mathbb{TL}}\), \(P_{\mathbb{TL}}\), and \(P_{loss}\) denote the costs associated with duplication, transfer, \(\mathbb{TL}\), \(\mathbb{TX}\), and loss events, respectively. The reconciliation cost of a DTLx reconciliation \(\alpha\) of \(G\) and \(S\) is then defined as follows. **Definition 7** (Reconciliation cost). : _Given a DTLx reconciliation \(\alpha=\langle\mathcal{L},G^{\prime},S^{\prime},\mathcal{M},\Sigma,\Delta,\Theta,\Theta _{L},\Theta_{X},\Xi,\tau\rangle\) for \(G\) and \(S\), the reconciliation cost associated with \(\alpha\) is the total cost of all events invoked by \(\alpha\).