Representing Asteroidal Sets on Subdivisions of Stars

Consider a simple undirected graph \( G = (V, E) \). A family of subtrees, \(\{T_v\}_{v \in V}\), of a tree \(\mathcal{T}\) is called a \((\mathcal{T}; t)\)-representation of \(G\) provided \( uv \in E \) if and only if \( |T_u \cap T_v| \geq t \). In this paper, we consider \((\mathcal{T}; t)\)-representations for graphs containing large asteroidal sets, where \(\mathcal{T}\) is a subdivision of the \(n\)-star \(K_{1, n}\). An asteroidal set in a graph \(G\) is a subset \(A\) of the vertex set such that for all 3-element subsets of \(A\), there exists a path in \(G\) between any two of these vertices which avoids the neighborhood of the third vertex. We construct a representation of an asteroidal set of size \( n + \sum_{k=2}^{n} \binom{n}{k} \binom{t-2}{k-1} \) and show that no graph containing a larger asteroidal set can be represented.