# Properties and Recognition of Atom Graphs
## 6 Atom Graph and Union Join Graph
As \(T(C_{2})\) is a subtree of \(T\), \(P_{T}(x,y_{j})\) is a path in \(T(C_{2})\), so \(i_{0}=min\{i\in[1,p]\), \(e_{i}\)_is an edge of_\(P_{T}(x,y_{j})\}\). As \(xy_{j}\) is an edge of \(G\), \(x\in V_{i_{0}}\). Hence, \(C\subseteq V_{i_{0}}\). As \(T(V_{i_{0}})\) is connected and each vertex of \(D_{2}\) is on the path in \(T\) between its two neighbors which belong to \(C\) and therefore to \(V_{i_{0}}\), \(C_{2}\subseteq V_{i_{0}}\). Let us show that for any triangle \((c,c^{\prime},d)\) in \(G\) such that \(d\in D_{1}\setminus V_{i_{0}}\), there is an integer \(j\) in \([1,p]\) such that \(V_{j}=e_{j}=c^{\prime}\). Let \((c,c^{\prime},d)\) be such a triangle. We assume w.l.o.g. that \(c\) is the neighbor of \(d\) in \(T\). Let \(i=min\{r\in[1,p]\), \(e_{i}\)_is an edge of_\(P_{T}(c^{\prime},d)\}\). As \(c^{\prime}d\) is an edge of \(G\), \(\{c^{\prime},d\}\subseteq V_{i}\). It follows that \(e_{i}=cd\) (otherwise \(e_{i}\) would be a subset of \(C_{2}\), and therefore an edge of \(T(V_{i_{0}})\), so \(V_{i}\) would be a subset of \(V_{i_{0}}\) and would not contain \(d\)). Hence, \(d\) is adjacent to each vertex of \(V_{i}\setminus\{d\}\), so \(V_{i}=\{c,c^{\prime},d\}\) and \(cc^{\prime}\) is an edge of \(T\). Let \(j\) be the integer such that \(cc^{\prime}=e_{j}\). As \(e_{j}\) is an edge of \(T(V_{i})\), \(V_{j}\subseteq V_{i}\) and as \(V_{j}\) contains the edge \(e_{j}\), but not the edge \(e_{i}\) since \(i<j\), \(V_{j}=e_{j}\). Now, as the vertices of \(D\) are pairwise non-adjacent in \(G\), there is some \(i\in\{1,2\}\) such that \(D\cap V_{i_{0}}\subseteq X_{i}\). We assume without loss of generality that \(i=1\). Then, \(D\cap V_{i_{0}}\subseteq V_{i_{0}}\cap X_{1}\subseteq N_{G}(y_{2})\). As \(G\) is a \(k\)-sun with \(k\geq 4\), there are \(2\) triangles \((c,c_{1},d_{1})\) and \((c,c_{2},d_{2})\) such that for each \(i\in\{1,2\}\), \(d_{i}\) is in \(D\setminus N_{G}(y_{2})\), and therefore in \(D\setminus V_{i_{0}}\), which is equal to \(D_{1}\setminus V_{i_{0}}\) since \(D_{2}\subseteq V_{i_{0}}\). So by the preceding argument, for each \(i\in\{1,2\}\) there is an integer \(j_{i}\) in \([1,p]\) such that \(V_{j_{i}}=e_{j_{i}}=cc_{i}\). Hence, there is no integer \(r\) such that \(e_{r}\) is an edge of \(P_{T}(c_{1},c_{2})\) and \(\{c_{1},c_{2}\}\subseteq V_{r}\), and therefore \(c_{1}c_{2}\) is not an edge of \(G\), a contradiction. We showed in Section 5 that by Characterization 6 the 3-sun is not a clique graph of a chordal graph. The same argument holds for each \(k\geq 3\): there is no integer \(k\) such that the \(k\)-sun is a clique graph of a chordal graph. However, the \(k\)-sun is a join-path-expanded tree for each \(k\geq 3\): it is easy to check that any spanning tree \(T\) of the \(k\)-sun which is a spanning tree of the union of its external triangles such that each vertex of degree \(2\) in the \(k\)-sun is of degree \(1\) in \(T\) is a join-path spanning tree of the \(k\)-sun. A join-path spanning tree of the \(4\)-sun is shown in Figure 8 (its edges are represented by full lines). It follows that the union of the classes of atom graphs and clique graphs of chordal graphs is a strict subclass of the class of join-path-expanded graphs.