Extremal trees with prescribed burning numbers

Graph burning is motivated by the spread of social influence, and the burning number measures the speed of the spread. A natural problem is to characterize graphs that are extremal with respect to order for a prescribed burning number. Since the burning number of a connected graph is attained by one of its spanning trees, the study of extremal trees provides insight into this problem. Accordingly, we focus on identifying such extremal trees, up to graph homeomorphism, for a prescribed burning number. In this work, we propose the concept of admissible sequences over a homeomorphically irreducible tree in addition to developing a general framework. We then determine whether an admissible sequence induces an extremal tree with a specified burning number. Additionally, we obtain some results on the smallest attainable diameter for extremal n -spiders with a prescribed burning number.