## 2 Preliminaries
### Distance-Hereditary Graphs  
The distance between two vertices in a graph is the length of the shortest path connecting them. This length is measured by the number of edges in the path. If no path exists between the two vertices, the distance is considered to be infinite. This concept is fundamental in graph theory and is used to determine how \"close\" or \"far apart\" vertices are within a graph. The diameter is a related concept, measuring the greatest distance between any two vertices. A graph is distance-hereditary if the distance between any two vertices in a connected induced subgraph is the same as in the original graph. Thus, any induced subgraph of a distance-hereditary graph is also distance-hereditary. Theorem 1 gives the recursive definition of distance-hereditary graphs. Theorem 1 ([20; 21]): _Distance-hereditary graphs can be defined recursively as follows:_  
1. _A graph_ \(G\) _consisting of a single vertex_ \(v\) _is distance-hereditary, and the twin set_ \(TS(G)=\{v\}\)_. Such a graph is called a primitive distance-hereditary graph;_
2. _Let_ \(G_{1}\) _and_ \(G_{2}\) _be two disjoint distance-hereditary graphs with the twin sets_ \(TS(G_{1})\) _and_ \(TS(G_{2})\)_, respectively. A graph_ \(G\) _with_ \(|V(G)|\geq 2\)_, obtained from_ \(G_{1}\) _and_ \(G_{2}\) _through one of the following three operations, is also a distance-hereditary graph:_ 1. _False twin operation: The graph_ \(G\) _is formed by the union of_ \(G_{1}\) _and_ \(G_{2}\)_, with_ \(TS(G)=TS(G_{1})\cup TS(G_{2})\)_, and denoted by_ \(G=G_{1}\bigodot G_{2}\)_;_ 2. _True twin operation: The graph_ \(G\) _is formed by connecting every vertex of_ \(TS(G_{1})\) _to all vertices of_ \(TS(G_{2})\)_, with_ \(TS(G)=TS(G_{1})\cup TS(G_{2})\)_, and denoted by_ \(G=G_{1}\bigotimes G_{2}\)_;_ 3. _Pendant vertex operation: The graph_ \(G\) _is formed by connecting every vertex of_ \(TS(G_{1})\) _to all vertices of_ \(TS(G_{2})\)_, with_ \(TS(G)=TS(G_{1})\)_, and denoted by_ \(G=G_{1}\bigoplus G_{2}\)_._  
By Theorem 1, a distance-hereditary graph \(G\) has a twin set \(TS(G)\), which is a subset of \(V(G)\) and defined recursively. This theorem also describes the construction of a binary ordered decomposition tree, known as a \(PTF\)-tree, which consists of \(2n-1\) nodes and can be built in linear time. In a \(PTF\)-tree, each leaf represents a single-vertex graph, and each internal node labeled by \(\bigoplus\), \(\bigotimes\), or \(\bigodot\) corresponds to an operation from the theorem. Figure 1 illustrates a distance-hereditary graph and its \(PTF\)-tree. Definition 5: Let \(C(G)\) denote the set of all maximal cliques in a distance-hereditary graph \(G\). Consequently, \(C(G[TS(G)])\) represents the set of all maximal cliques in the subgraph \(G[TS(G)]\). We define \(C_{T}(G)\) as the set of maximal cliques in \(G\) that are also maximal cliques in \(G[TS(G)]\). Conversely, \(C_{\overline{T}}(G)\) is defined as the set of maximal cliques in \(G\) that are not maximal cliques in \(G[TS(G)]\). Thus, the set of all maximal cliques in \(G\) can be expressed as \(C(G)=C_{T}(G)\cup C_{\overline{T}}(G)\). Definition 6: Let \(G_{1}\) and \(G_{2}\) be disjoint distance-hereditary graphs. Suppose that \(G\) is either \(G=G_{1}\bigoplus G_{2}\) or \(G=G_{1}\bigotimes G_{2}\). We define \(C_{X}(G)\) as the set \(\{C_{1}\cup C_{2}\mid C_{1}\in C(G_{1}[TS(G_{1})])\), and \(C_{2}\in C(G_{2}[TS(G_{2})])\}\). Figure 1: (a) A distance-hereditary graph \(G\). (b) A \(PTF\)-tree of \(G\). **Lemma 1** ([22]). : _Suppose that \(G\) is a primitive distance-hereditary graph with the vertex \(v\). Then, \(C(G)=C(G[TS(G))])=C_{T}(G)\) and \(C_{\overline{T}}(G)=\emptyset\)._  
**Lemma 2** ([22]). : _Let \(G_{1}\) and \(G_{2}\) be disjoint distance-hereditary graphs. If \(G=G_{1}\bigodot G_{2}\), then_  
1. \(C(G[TS(G)])=C(G_{1}[TS(G_{1})])\cup C(G_{2}[TS(G_{2})])\)_;_
2. \(C(G)=C(G_{1})\cup C(G_{2})\)_;_
3. \(C_{T}(G)=C_{T}(G_{1})\cup C_{T}(G_{2})\)_; and_
4. \(C_{\overline{T}}(G)=C_{\overline{T}}(G_{1})\cup C_{\overline{T}}(G_{2})\)_._  
**Lemma 3** ([22]).