On Balance Index Sets of Halin graphs of Stars and Double Stars

Let \( G \) be a graph with vertex set \( V(G) \) and edge set \( E(G) \), and let \( A = \{0,1\} \). A labeling \( f: V(G) \to A \) induces a partial edge labeling \( f^*: E(G) \to A \) defined by \( f^*((u, v)) = f(u) \) if and only if \( f(u) = f(v) \) for each edge \( (u, v) \in E(G) \). For \( i \in A \), let \( \text{v}_f(i) = \text{card} \{v \in V(G) : f(v) = i\} \) and \( \text{e}_f(i) = \text{card} \{e \in E(G) : f^*(e) = i\} \). A labeling \( f \) of \( G \) is said to be friendly if \( |\text{v}_f(0) – \text{v}_f(1)| \leq 1 \). The \textbf{balance index set} of the graph \( G \), \( \text{BI}(G) \), is defined as \( \{|\text{e}_f(0) – \text{e}_f(1)| : \text{the vertex labeling } f \text{ is friendly}\} \). We determine the balance index sets of Halin graphs of stars and double stars.