# Properties and Recognition of Atom Graphs
## 4 Atom Graphs Are Perfect
Thus, \(\mu\) satisfies the first alternative of Property 9 and for each set of consecutive edges \(XY\) and \(YZ\) of \(\mu\) associated with the same clique minimal separator \(S\), \(C_{S}(X)=C_{S}(Z)\). _Second case_ : \(C_{S}(D)\
eq C_{S}(B)\)  
Let us show that for each node \(X\) of \(\mu-\{A,B,A^{\prime}\}\), \(S\
ot\subseteq X\). Let \(X\) be a node of \(\mu-\{A,B,A^{\prime}\}\), and let \(Y\) be a node of \(\{A,B\}\) such that \(C_{S}(X)\
eq C_{S}(Y)\) and \(XY\) is not an edge of \(\mu\) (if \(X=D\) then \(Y=B\) otherwise \(Y\) exists since neither \(XA\) nor \(XB\) is an edge of \(\mu\) and \(C_{S}(A)\
eq C_{S}(B)\)). By \(P_{1}\)\(S\
ot\subseteq X\cap Y\), so as \(S\subseteq Y\), \(S\
ot\subseteq X\). It follows that for each edge \(XY\) of \(\mu-\{B\}\), \(S\
ot\subseteq X\cap Y\). Hence, \(S\) is associated with no other edge than the two consecutive edges \(AB\) and \(A^{\prime}B\) of \(\mu\). It also follows that \(S\) is a strict subset of no other clique minimal separator associated with an edge of \(\mu\), and as this holds for each inclusion-minimal clique minimal separator associated with an edge of \(\mu\), the clique minimal separators associated with the edges of \(\mu\) are pairwise non-inclusive and therefore all inclusion-minimal. Hence, \(\mu\) satisfies the second alternative of Property 9, and for each set of consecutive edges \(XY\) and \(YZ\) of \(\mu\) associated with the same clique minimal separator \(S\), \(C_{S}(X)=C_{S}(Z)\). **Example 1**. : _A graph and its atom graph having an induced \(C_{4}\) satisfying the first alternative of Property 9 is shown in Figure 3 (the cycle \((A,D,C,F)\) is such a \(C_{4}\)). A graph and its atom graph having an induced \(C_{4}\) satisfying the second alternative of Property 9 is shown in Figure 4 (the cycle \((A,B,E,D)\) is such a \(C_{4}\)). Note that in both cases, the graph is chordal and has a clique tree which is a path: the path \((A,B,C,D,E,F)\), for instance, in the first case, and \((A,B,C,D,E)\), for instance, in the second case. This contradicts the claim from [18] that any path of a clique tree of a chordal graph induces a chordal subgraph of its atom graph._  
_For each \(k\geq 3\), the atom graph of the graph obtained from a \(k\)-sun by adding a vertex of degree \(1\) adjacent to \(x\) for each vertex \(x\) of the clique of size \(k\) has an induced \(C_{2k}\) satisfying the second alternative of Property 9 (see Figure 2 for \(k=3\))._  
**Corollary 1**: _Each atom graph is odd-hole-free._  
**Property 10**: _Each atom graph is anti-hole-free._  
We assume for contradiction that some induced subgraph of some atom graph \(G\) is an anti-hole and let \(k\) be its length. \(k\geq 6\) by Corollary 1 since an anti-hole of length \(5\) is a hole. Let \(G^{\prime}\) be a graph such that \(G\) is isomorphic to \(AG(G^{\prime})\), let \(\mu\) be a chordless cycle of \(\overline{AG(G^{\prime})}\) of length \(k\), and let \((A^{\prime},A,D_{1},B_{1},B_{2})\) be a sequence of consecutive nodes of \(\mu\). As \(k\geq 6\)\(\{A^{\prime},A,B_{1},B_{2}\}\) induces a \(C_{4}\) of \(AG(G^{\prime})\), so by Property 9\(A\cap B_{1}\) is equal to \(A^{\prime}\cap B_{1}\) or to \(A\cap B_{2}\). We assume without loss of generality that \(A\cap B_{1}=A^{\prime}\cap B_{1}\). Let \((A^{\prime},A,D_{1},B_{1},\ldots,B_{p},D_{p})\) be the full sequence of consecutive vertices of \(\mu\), and for each \(i\in[1,p]\), let \(S_{i}=A\cap B_{i}\) and let \(P(i)\) be the predicate \(S_{i}=A^{\prime}\cap B_{i}\). As \(P(1)\) holds and for each \(i\in[1,p-1]\),\(P(i)\Rightarrow P(i+1)\) holds by Property 9, it follows by induction that \(\forall i\in[1,p]\)\(P(i)\). Let \(Y\in\{D_{1},D_{p}\}\) and let \(i\in\{1,p\}\). As \(AB_{i}\) is an edge of \(AG(G^{\prime})\)\(C_{S_{i}}(A)\
eq C_{S_{i}}(B_{i})\), and by Property 9\(C_{S_{i}}(A)=C_{S_{i}}(A^{\prime})\). By Lemma 3 c) with \(\{X,X^{\prime}\}=\{A,A^{\prime}\}\), \(C_{S_{i}}(A)=C_{S_{i}}(Y)\).