Given a graph G, let μ(G) denote the size of the smallest maximal independent set of G. A family of sets is called a star if some element is in every set of the family. A split vertex has degree at least 3. Holroyd and Talbot conjectured the following Erdős-Ko-Rado-type statement about intersecting families of independent sets of graphs: if 1≤r≤μ(G)/2 then there is an intersecting family of independent r-sets of maximum size that is a star. In this paper we prove similar statements for sparse graphs on n vertices: roughly, for graphs of bounded average degree with r≤O(n1/3), for graphs of bounded degree with r≤O(n1/2), and for trees having a bounded number of split vertices with r≤O(n1/2).