The k -leaf power graph G of a tree T is a graph whose vertices are the leaves of T and whose edges connect pairs of leaves at unweighted distance at most k in T . Recognition of the k -leaf power graphs for $$k \ge 7$$ k ≥ 7 is still an open problem. In this paper, we provide two algorithms for this problem for sparse leaf power graphs. Our results shows that the problem of recognizing these graphs is fixed-parameter tractable when parameterized both by k and by the degeneracy of the given graph. To prove this, we first describe how to embed a leaf root of a leaf power graph into a product of the graph with a cycle graph. We bound the treewidth of the resulting product in terms of k and the degeneracy of G . The first presented algorithm uses methods based on monadic second-order logic ( $${\text{MSO}}_2$$ MSO 2 ) to recognize the existence of a leaf power as a subgraph of the graph product. Using the same embedding in the graph product, the second algorithm presents a dynamic programming approach to solve the problem and provide a better dependence on the parameters.

中文翻译：

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通过嵌入到图形产品中的参数化叶功率识别

树 T 的 k 叶幂图 G 是一个图，其顶点是 T 的叶子，其边连接 T 中最多 k 的未加权距离的叶子对。识别 $$k \ge 7$$ k ≥ 7 的 k 叶功率图仍然是一个悬而未决的问题。在本文中，我们为稀疏叶功率图的这个问题提供了两种算法。我们的结果表明，当通过 k 和给定图的简并性进行参数化时，识别这些图的问题是固定参数易于处理的。为了证明这一点，我们首先描述如何将叶功率图的叶根嵌入到该图与循环图的乘积中。我们根据 k 和 G 的简并性来限制结果乘积的树宽。第一个提出的算法使用基于一元二阶逻辑 ($${\text{MSO}}_2$$ MSO 2 ) 的方法来识别叶功率的存在作为图产品的子图。在图产品中使用相同的嵌入，第二个算法提出了一种动态规划方法来解决问题并提供更好的参数依赖性。