Given a graph G and an integer r ≥ 1, let I (r) (G) denote the family of independent sets of size r of G. For a vertex v of G, let I (r) v (G) denote the family of independent sets of size r that contain v. This family is called an r-star and v is the centre of the star. Then G is said to be r-EKR if no pairwise intersecting subfamily of I (r) (G) is bigger than the largest r-star, and if every maximum size pairwise intersecting subfamily of I (r) (G) is an r-star, then G is said to be strictly r-EKR. Let µ(G) denote the minimum size of a maximal independent set of G. Holroyd and Talbot conjectured that if 2r ≤ µ(G), then G is r-EKR and strictly r-EKR if 2r < µ(G).An elongated claw is a tree in which one vertex is designated the root and no vertex other than the root has degree greater than 2. A depth-two claw is an elongated claw in which every vertex of degree 1 is at distance 2 from the root. We show that if G is a depth-two claw, then G is strictly r-EKR if 2r ≤ µ(G) + 1, confirming the conjecture of Holroyd and Talbot for this family. We also show that if G is an elongated claw with n leaves and at least one leaf adjacent to the root, then G is r-EKR if 2r ≤ n.Hurlbert and Kamat had conjectured that one can always find a largest r-star of a tree whose centre is a leaf. Baber and Borg have separately shown this to be false. We show that, moreover, for all n ≥ 2, d ≥ 3, there exists a positive integer r such that there is a tree where the centre of the largest r-star is a vertex of degree n at distance at least d from every leaf.
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