In this paper, we introduce the problem of finding an orientation of a given undirected graph that maximizes the burning number of the resulting directed graph. We show that the problem is polynomial-time solvable on K\H{o}nig-Egerv\'{a}ry graphs (and thus on bipartite graphs) and that an almost optimal solution can be computed in polynomial time for perfect graphs. On the other hand, we show that the problem is NP-hard in general and W[1]-hard parameterized by the target burning number. The hardness results are complemented by several fixed-parameter tractable results parameterized by structural parameters. Our main result in this direction shows that the problem is fixed-parameter tractable parameterized by cluster vertex deletion number plus clique number (and thus also by vertex cover number).

翻译：本文引入了一个问题：给定无向图，寻找其一个定向，使得所得有向图的燃烧数最大化。我们证明该问题在König-Egerváry图（从而在二分图）上可以在多项式时间内求解，并且在完美图上可以在多项式时间内计算出近似最优解。另一方面，我们证明该问题在一般情况下是NP难的，且以目标燃烧数为参数时是W[1]-难的。我们还通过若干以结构参数为参数的固定参数可解结果对上述硬度结果进行了补充。本方向的主要结果表明，该问题在以团顶点删除数加团数（从而也以顶点覆盖数）为参数时是固定参数可解的。