# Exploring Graph and Digraph Persistence
## 4 Simple Features  
We now define a class of features extending the class of monotone features (Definition 11). **Definition 12**. : _Let \(G=(V,E)\)  
\(H_{2}=G_{\bar{u}}\), \(H_{3}=G_{\bar{u}}\). Then, by Definition 12, \(\mathcal{F}(X)=false\) in \(G_{w^{\prime}}\). Thus, between two values for which \(\mathcal{F}(X)=true\), there cannot be a value such that \(\mathcal{F}(X)=false\). The interval \([u_{1},v_{1})\) of Lemma 13, i.e., the widest interval for which \(\mathcal{F}(X)=true\) in \((G,f)\), is called the \(\mathcal{F}\)-interval of \(X\) in \((G,f)\). Proposition 5: \(\sigma^{\mathcal{F}}=\varrho^{\mathcal{F}}\). Proof: By Lemma 13, \(\sigma^{\mathcal{F}}\) and \(\varrho^{\mathcal{F}}\) would differ only if, for at least one set \(X\), there were values \(w_{1}<w_{2}<w_{3}\) for which \(\mathcal{F}(X)=true\) in \(G_{w_{1}}\) and in \(G_{w_{3}}\) but \(\mathcal{F}(X)=false\) in \(G_{w_{2}}\). If course, there exist features that are not simple; as an example, for a (di)graph \(G\), \(\mathcal{E}U(X)=true\) if and only if \(X\) is a set of vertices inducing a nonempty Eulerian sub(di)graph, neither \(\mathcal{E}U\) nor its maximal version \(\mathcal{E}U\) (simply \(\mathcal{E}U\) in [8]) is simple, as can be seen in [8] (Fig. 2), depicting a graph with \(\sigma^{m\mathcal{E}U}\
eq\varrho^{m\mathcal{E}U}\). It is easy to prove that the following features are simple. For a graph \(G\), the (rather trivial) feature \(\mathcal{D}\varrho^{d}\) for which \(\mathcal{D}\varrho^{d}(X)=true\) if and only if \(X\) is a singleton containing a vertex of a fixed degree \(d\) is simple. Also, the feature \(\mathcal{D}\varrho^{\geq d}\) for which \(\mathcal{D}\varrho^{\geq d}(X)=true\) if and only if \(X\) is a singleton containing a vertex whose degree is \(\geq d\) is simple, with half-lines as \(\mathcal{D}\varrho^{\geq d}\) intervals. Analogous features are defined for a digraph in terms of indegree and outdegree. For a (di)graph \(G\), \(\mathcal{C}om(X)=true\) if and only if \(X\) is a set of vertices inducing a connected subgraph (resp., a strongly connected subdigraph) of \(G\). Conn is simple, and its Conn-intervals are half-lines. For a (di)graph \(G\), \(\mathcal{C}om(X)=true\) if and only if \(X\) is a set of vertices inducing a connected component (resp., a strong component) of \(G\). Comp is simple; for the Comp-interval \([u_{1},v_{1})\) of a set \(X\), the value \(v_{1}\) is either \(+\infty\) or the value at which some other vertex joins the component (resp., strong component) induced by \(X\). For a (di)graph \(G\), any monotone feature \(\mathcal{F}\) is simple. Proof: Let \(G=(V,E)\) be a (di)graph and \(\mathcal{F}\) be a monotone feature. For \(H_{1}=(V_{1},E_{1})\subseteq H_{2}=(V_{2},E_{2})\subset H_{3}\subseteq G\) and \(X\subseteq(V\cup E)\), assume that \(\mathcal{F}(X)=true\) in \(H_{3}\) and \(\mathcal{F}(X)=false\) in \(H_{2}\). By condition 1 of Definition 11, \(X\
subseteq V_{2}\cup E_{2}\) (recall the convention of Definition 6), so \(X\
subseteq V_{1}\cup E_{1}\) and \(\mathcal{F}(X)=false\) in \(H_{1}\). Let \(f,g:E\rightarrow\mathbb{R}\) be filtering functions on a (di)graph \(G\), and assume that there exists a positive real number \(h\) such that \(\sup_{e\in E}\lvert f(e)-g(e)\rvert\leq h\). Then, for real numbers \(\bar{u},\bar{v}\) such that \(\bar{u}<\bar{v}-h\), the sublevel (di)graph \(G_{f,\bar{u}}\) is a sub(di)graph of \(G_{g,\bar{v}}\), and \(G_{g,\bar{u}}\) is a sub(di)graph of \(G_{f,\bar{v}}\). Proof: For any \(e\in E\), if \(f(e)\leq\bar{u}\), then \(g(e)\leq\bar{u}+h<\bar{v}\); analogously, if \(g(e)\leq\bar{u}\), then \(f(e)\leq\bar{u}+h<\bar{v}\). The following Proposition generalizes Proposition 4 (i.e., [8] (Prop. 4)) with essentially the same proof. The \(ip\)-function generators \(\sigma^{\mathcal{F}}=\varrho^{\mathcal{F}}\) are balanced, and hence stable. Proof: With the same hypotheses as in Lemma 2, let \(X\subseteq(V\cup E)\) be such that \(\mathcal{F}(X)=true\) in \(G_{f,\bar{w}}\) at all levels \(w\in[u-h,v+h]\), with \(u<v\).