Burnability of Double Spiders and Path Forests

The burning number of a graph can be used to measure the spreading speed of contagion in a network. The burning number conjecture is arguably the main unresolved conjecture related to this graph parameter, which can be settled by showing that every tree of order $m^2$ has burning number at most $m$. This is known to hold for many classes of trees, including spiders - trees with exactly one vertex of degree greater than two. In fact, it has been verified that certain spiders of order slightly larger than $m^2$ also have burning numbers at most $m$, a result that has then been conjectured to be true for all trees. The first focus of this paper is to verify this slightly stronger conjecture for double spiders - trees with two vertices of degrees at least three and they are adjacent. Our other focus concerns the burning numbers of path forests, a class of graphs in which their burning numbers are naturally related to that of spiders and double spiders. Here, our main result shows that a path forest of order $m^2$ with a sufficiently long shortest path has burning number exactly $m$, the smallest possible for any path forest of the same order.