## 2 Preliminaries  
Throughout this paper, we only consider undirected simple graphs (unless otherwise stated). Let \(G\) be a graph. We use \(n\) to denote the number of vertices in \(G\). We denote the sets of vertices and edges in \(G\) with \(V(G)\) and \(E(G)\), respectively. For \(v\in V(G)\), the set of neighbors of \(v\) in \(G\) is denoted by \(N_{G}(v)\). This notation is extended to vertex sets: \(N_{G}(X)=\bigcup_{v\in X}N_{G}(v)\setminus X\) for \(X\subseteq V(G)\). For \(X\subseteq V(G)\), the subgraph of \(G\) induced by \(X\) is denoted by \(G[X]\). We define \(N_{G}[v]=N_{G}(v)\cup\{v\}\) and \(N_{G}[X]=N_{G}(X)\cup X\) for \(v\in V(G)\) and \(X\subseteq V(G)\). When the subscripts are clear from the context, we may omit them. For \(v\in V(G)\), we write \(G-v\) to denote the graph obtained from \(G\) by deleting \(v\) (and its incident edges). A set of vertices \(C\) is called a _vertex cover_ of \(G\) if for every edge in \(G\), at least one end vertex of it belongs to \(C\). A vertex cover \(C\) is said to be _connected_ if \(G[C]\) is connected. A _dominating set_ of \(G\) is a set of vertices \(D\) such that \(V(G)=N[D]\), that is, every vertex in \(V(G)\setminus D\) has a neighbor in \(D\). A connected dominating set of \(G\) is defined analogously. The \"capacitated\" variants of a vertex cover and dominating set are defined as follows. Let \(c\colon V(G)\to\mathbb{N}\) be a capacity function. Notice that we allow a vertex \(v\) such that \(c(v)=0\). A _capacitated vertex cover_ of \((G,c)\) is a pair \((C,\alpha)\) of a vertex set \(C\subseteq V(G)\) and a function \(\alpha\colon E(G)\to C\) such that \(\alpha(\{u,v\})\) is either \(u\) or \(v\) and \(|\alpha^{-1}(v)|\leq c(v)\) for \(v\in C\), where \(\alpha^{-1}(v)=\{e\in E(G):\alpha(e)=v\}\) is the set of edges mapped to \(v\) under \(\alpha\). In other words, a vertex \(v\in C\) covers at most \(c(v)\) edges. A _capacitated dominating set_ of \((G,c)\) is a pair \((D,\beta)\) of a vertex set \(D\) and a function \(\beta\colon V(G)\setminus D\to D\) such that \(\beta(v)\) is a neighbor of \(v\) and \(|\beta^{-1}(v)|\leq c(v)\) for \(v\in D\), where \(\beta^{-1}(v)=\{w\in V(G)\setminus D:\beta(w)=v\}\). We simply refer to a vertex set \(X\subseteq V(G)\) as a capacitated vertex cover (resp., a capacitated dominating set) of \((G,c)\) if there is a function \(\alpha\colon E(G)\to X\) (resp., \(\beta\colon V(G)\setminus X\to X\)) such that \((X,\alpha)\) is a capacitated vertex cover (resp.,\((X,\beta)\) is a capacitated dominating set) of \((G,c)\). Clearly, every capacitated vertex cover (resp., a capacitated dominating set) of \((G,c)\) is a vertex cover (resp., a dominating set) of \(G\), and the converse holds when setting \(c(v)=\Delta\) for \(v\in V(G)\), where \(\Delta\) is the maximum degree of a vertex in \(G\). Let \(\mathcal{S}\subseteq 2^{V(G)}\) be a collection of vertex sets of \(G\). We say that \(\mathcal{S}\) is _monotone_ if for \(X,Y\subseteq V(G)\) with \(X\subseteq Y\), \(X\in\mathcal{S}\) implies \(Y\in\mathcal{S}\). It is easy to see that the collections of capacitated vertex covers and capacitated dominating sets of \((G,c)\) are monotone, while the collection of connected vertex sets (i.e., vertex sets that induce connected subgraphs of \(G\)) is not monotone. The following proposition shows that the collections of connected vertex covers and connected dominating sets of \(G\) are also monotone. Proposition 1: _Let \(G=(V,E)\) be a connected graph and let \(X\) be a connected vertex cover (resp., a connected dominating set) of \(G\). Then, for any \(Y\subseteq V(G)\) with \(X\subseteq Y\), \(Y\) is a connected vertex cover (resp., a connected dominating set) of \(G\) as well._  
Proof: Let \(X\) be a connected vertex cover of \(G\). We assume that \(G\) has at least two vertices, as otherwise the lemma is trivial. Moreover, we assume that \(X\) has at least one vertex. As \(G\) is connected and \(X\) is a vertex cover of \(G\), every vertex \(v\) in \(V(G)\setminus X\) has a neighbor in \(X\).