Subdivided trees are integral sum graphs

A graph image is an integral sum graph (ISG) if there exists a labeling image such that image and for every pair of distinct vertices image, image is an edge if and only if image. A vertex in a graph is called a fork if its degree is not 2. In 1998, Chen proved that every tree whose forks are at distance at least 4 from each other is an ISG. In 2004, He et al. reduced the distance to 3. In this paper we reduce the distance further to 2, i.e. we prove that every tree whose forks are at least distance 2 apart is an ISG.