# Properties and Recognition of Atom Graphs
## 2 Preliminaries  
The preliminaries of this paper are similar to those of [21]. In order to avoid useless repetition, we only recall here the main definitions, results and notations, and add some results from [21], the new notions appearing in this paper and the results that are explicitly referred to in this paper. We invite the reader to have a look at the preliminaries section of [21] for more detail and an example. We consider here finite and undirected graphs. For a graph \(G=(V,E)\), \(n=|V|\) (order of \(G\)) and \(m=|E|\). For any subset \(X\) of \(V\), \(G(X)\) denotes the subgraph of \(G\) induced by \(S\). For any vertex \(v\) of \(G\), \(N_{G}(v)\) denotes the neighborhood of \(v\) in \(G\), i.e., \(N_{G}(v)=\{w\in V\mid vw\in E\}\), and \(N_{G}[v]\) denotes its closed neighborhood in \(G\), i.e., \(N_{G}[v]=\{v\}\cup N_{G}(v)\). For any subset \(X\) of \(V\), \(N_{G}(X)=(\cup_{v\in X}N_{G}(v))\setminus X\) and \(N_{G}[X]=X\cup N_{G}(X)\). For each graph \(G\), \(K(G)\) denotes the set of maximal cliques of \(G\) and the _clique graph_ of \(G\), denoted by \(CG(G)\), is the intersection graph of \(K(G)\). If \(X\) and \(Y\) are nodes of a tree \(T\), \(P_{T}(X,Y)\) denotes the path in \(T\) between \(X\) and \(Y\). For each integer \(k\geq 3\), a \(C_{k}\) is a graph which is a cycle of length \(k\) and a \(k\)-_sun_ is a graph obtained from a \(C_{k}\) by adding for each edge \(xy\) of this \(C_{k}\) a vertex adjacent to \(x\) and \(y\) and adding all edges that are necessary to make this \(C_{k}\) into a clique. A _hole_ is a \(C_{k}\) with \(k\geq 5\), and an _anti-hole_ is its complement graph. A hole or anti-hole is _odd_ if its number of vertices is odd. \(G\) is _(odd-)(anti-)hole-free_ if no induced subgraph of \(G\) is an (odd)(anti-)hole. We do not use the definition of a _perfect_ graph in this paper, but we use its well-known characterization as an odd-hole-free and odd-anti-hole-free graph (Strong Perfect Graph Theorem). _Separation._ Let \(G=(V,E)\) be a connected graph and let \(S\) be a subset of \(V\). \(S\) is a _separator_ of \(G\) if \(G(V\setminus S)\) is disconnected. The connected components of \(G(V\setminus S)\) are called the _components of_\(S\)_in \(G\). For any pair \(\{a,b\}\) of \(V\setminus S\), \(S\) is an _ab-separator_ of \(G\) if \(a\) and \(b\) are in different components of \(S\) in \(G\). \(S\) is a _minimal ab-separator_ if it is an inclusion-minimal _ab_-separator, and a _minimal separator_ if there is some pair \(\{a,b\}\) of \(V\) such that \(S\) is a minimal _ab_-separator. A _full component_ of \(S\) in \(G\) is a component \(C\) of \(S\) in \(G\) such that \(N_{G}(C)=S\). \(S\) is a minimal separator if \(S\) has at least two full components, and \(S\) is a minimal ab-separator if \(a\) and \(b\) lie in two different full components of \(S\). Let \(A\) and \(B\) be two subsets of \(V\). \(S\) is a (minimal) \(AB\)-separator of \(G\) if it is a (minimal) _ab_-separator of \(G\) for each \(a\in A\) and each \(b\in B\). If \(G\) is disconnected, then a (minimal) (_ab_-)separator of \(G\) is a (minimal) (_ab_-)separator of one of its connected components. _Chordal graph._ A graph is _chordal_, or _triangulated_, if it has no chordless cycle of length at least 4. A graph is chordal if and only if all its minimal separators are cliques [22]. A connected graph is chordal if and only if it has a _clique tree_[16; 23]. **Definition 1**. : _Let \(G=(V,E)\) be a connected chordal graph. A clique tree of \(G\) is a tree \(T=(\mathcal{K}(G),E_{T})\) such that for each vertex \(x\) of \(G\), the set \(\mathcal{K}_{x}\) of nodes of \(T\) containing \(x\) induces a subtree of \(T\)._  
**Characterization 1** ([15]). : _Let \(G=(V,E)\) be a connected chordal graph, let \(T\) be a clique tree of \(G\), and let \(S\subseteq V\); then \(S\) is a minimal separator of \(G\) if and only if there is an edge \(XY\) of \(T\) such that \(S=X\cap Y\)._  
**Property 1** ([24]).