# Properties and Recognition of Atom Graphs
## 2 Preliminaries
: _A graph \(G\) is a clique graph of a chordal graph if and only if it has a spanning tree \(T\) such that each maximal clique of \(G\) induces a subtree of \(T\)._  
\(\alpha\)_-acyclic hypergraphs._ A _simple hypergraph_, or _hypergraph_ for short, is a structure \(H=(V,\mathcal{E})\), where \(V\) is its vertex set and \(\mathcal{E}\) is a set of non-empty subsets of \(V\), called the _hyperedges_ of \(H\), whose union is equal to \(V\). A hypergraph is a _clutter_ if the elements of \(\mathcal{E}\) are pairwise non-inclusive. Its _line graph_, denoted by \(L(H)\), is the intersection graph of \(\mathcal{E}\). Its _2-section graph_, denoted by \(2 SEC(H)\), is the graph whose vertex set is \(V\) and whose edges are the pairs of \(V\) that are contained in a hyperedge of \(H\). \(H\) is _connected_ if \(L(H)\) is connected, or equivalently if \(2 SEC(H)\) is connected. A _join tree_ of \(H\) is a tree \(T\) whose node set is \(\mathcal{E}\) and such that for each vertex \(x\) of \(H\), the set \(\mathcal{E}_{x}\) of nodes of \(T\) containing \(x\) induces a subtree of \(T\), or equivalently, such that for each pair \(\{X,Y\}\) of \(\mathcal{E}\), \(X\cap Y\) is a subset of each node of \(P_{T}(X,Y)\). \(H\) is \(\alpha\)_-acyclic_ if it has a join tree. Definition 4 ([21]). : _The union join graph of an \(\alpha\)-acyclic hypergraph \(H\), denoted by \(UJ(H)\), is the union of its join trees._  
Definition 5 ([21]). : _Let \(G=(V,E)\) be a graph. The atom hypergraph of \(G\) is the hypergraph \(H=(V,\mathcal{A}(G))\)._  
Property 4 ([21]). : _Let \(G\) be a connected graph and let \(H\) be its atom hypergraph. Then, \(H\) is a connected \(\alpha\)-acyclic hypergraph and the atom graph of \(G\) is the union join graph of \(H\)._  
Property 5.: _Let \(H\) be an \(\alpha\)-acyclic hypergraph, and let \(G\) be the 2-section graph \(2 SEC(H)\). Then, \(G\) is chordal and if \(H\) is a clutter then it is the atom hypergraph of \(G\)._  
A non-clutter \(\alpha\)-acyclic hypergraph can be turned into a clutter by adding a specific element to each non-inclusion-maximal hyperedge, so any connected \(\alpha\)-acyclic hypergraph is an atom hypergraph up to some isomorphism as stated in Property 6. Property 6 ([21]). : _Let \(H\) be a connected \(\alpha\)-acyclic hypergraph. Then, there is a connected chordal graph \(G\) such that the atom graph of \(G\) is isomorphic to the union join graph of \(H\)._  
Characterization 5 can be rewritten in terms of \(\alpha\)-acyclic hypergraph as follows (_tuj_ stands for _to union join_). Definition 6 ([21]). : _For each join tree \(T=(\mathcal{E},E_{T})\) of a hypergraph, \(tuj(T)\) is the graph whose node set is \(\mathcal{E}\) and whose edges are the pairs \(\{X,Y\}\) of \(\mathcal{E}\) such that there is an edge \(X^{\prime}Y^{\prime}\) of \(P_{T}(X,Y)\) such that \(X\cap Y=X^{\prime}\cap Y^{\prime}\) (or equivalently \(X^{\prime}\cap Y^{\prime}\subseteq X\cap Y\))._  
Characterization 7 ([21]). : _For each \(\alpha\)-acyclic hypergraph \(H\) and each join tree \(T\) of \(H\), \(UJ(H)=tuj(T)\)._  
We associate with each edge \(AB\) of a join tree or an union join graph the set \(A\cap B\) of weight \(|A\cap B|\).