Induced forests in some Distance Regular Graphs


The celebrated Erdos-Ko-Rado theorem can be interpreted as a characterization of the size and structure of independent sets/co-cliques of maximum size in Kneser Graphs. Similar characterizations have been observed in a few other classes of Distance Regular Graphs. In this talk, we will consider a closely related problem of finding the size and structure of the biggest induced forest in a graph. This question is inspired by a graph process known as `Bootstrap Percolation’. In this process, given a fixed threshold $r$ and a set of ‘initially infected’ vertices, the states of vertices (either healthy or infected) are updated indiscrete time steps: any healthy vertex with at least $r$ infected neighbours becomes itself infected. An initially infected set of vertices percolates if the infection spreads to the entire graph. One of the extremal questions related to this process is to find the smallest percolating set. In the special case of $d$-regular graphs, when $r=d$, a set percolates exactly when its complement is independent. Meanwhile, in the case $r=d−1$, a set percolates if its complement is a set of vertices that induce a forest. We will characterize maximum induced forests in the Kneser graph and some other families of Distance Regular Graphs.