Since counting subgraphs in general graphs is, by and large, a computationally demanding problem, it is natural to try and design fast algorithms for restricted families of graphs. One such family that has been extensively studied is that of graphs of bounded degeneracy (e.g., planar graphs). This line of work, which started in the early 80’s, culminated in a recent work of Gishboliner et al., which highlighted the importance of the task of counting homomorphic copies of cycles (i.e., cyclic walks) in graphs of bounded degeneracy.

Our main result in this paper is a surprisingly tight relation between the above task and the well-studied problem of detecting (standard) copies of directed cycles in general directed graphs. More precisely, we prove the following:

• One can compute the number of homomorphic copies of C_2k and C_2k+1 in n-vertex graphs of bounded degeneracy in time Õ(n^d_k), where the fastest known algorithm for detecting directed copies of C_k in general m-edge digraphs runs in time Õ(m^d_k).

• Conversely, one can transform any O(n^b_k) algorithm for computing the number of homomorphic copies of C_2k or of C_2k+1 in n-vertex graphs of bounded degeneracy, into an Õ(m^b_k) time algorithm for detecting directed copies of C_k in general m-edge digraphs.

We emphasize that our first result does not use a black-box reduction (as opposed to the second result which does). Instead, we design an algorithm for computing the number of C_k-homomorphisms in degenerate graphs and show that one part of its analysis can be reduced to the analysis of the fastest known algorithm for detecting directed cycles in general digraphs, which was carried out in a recent breakthrough of Dalirrooyfard, Vuong and Vassilevska Williams. As a by-product of our algorithm, we obtain a new algorithm for detecting k-cycles in directed and undirected graphs of bounded degeneracy that is faster than all previously known algorithms for 7 ≤ k ≤ 11, and faster for all k ≥ 7 if the matrix multiplication exponent is 2.

由于计算一般图中的子图总的来说是一个计算要求很高的问题，因此尝试为受限图族设计快速算法是很自然的。已被广泛研究的此类家族之一是有界退化图（例如，平面图）。这一系列工作始于 80 年代初，在 Gishboliner 等人最近的一项工作中达到顶峰，该工作强调了计算有界退化图中循环的同态副本（即循环游走）任务的重要性。

我们在本文中的主要结果是上述任务与在一般有向图中检测有向循环的（标准）副本的经过充分研究的问题之间存在惊人的紧密关系。更准确地说，我们证明了以下内容：

• 可以计算C _2k和C _2k+1在时间 Õ( n ^{d _k} ) 的有界退化的n顶点图中C 2k 和C 2k+1 的同态副本的数量，其中已知最快的算法用于在一般m边上检测C _k的有向副本digraphs 在时间 Õ( m ^{d _k} )中运行。

• 相反，可以将任何用于计算C _2k或C _2k+1在有界退化的n顶点图中的同态副本数的O(n ^{b _k} )算法转换为用于检测的 Õ( m ^{b _k} ) 时间算法一般m边有向图中C _k的有向副本。

我们强调我们的第一个结果没有使用黑盒缩减（与第二个结果相反）。相反，我们设计了一种算法来计算退化图中C _k -同态的数量，并表明其分析的一部分可以简化为对检测一般有向图中有向循环的已知最快算法的分析，该算法在Dalirrooyfard、Vuong 和 Vassilevska Williams 最近的突破。作为我们算法的副产品，我们获得了一种新算法，用于检测有界退化的有向和无向图中的k循环，该算法比所有先前已知的 7 ≤ k ≤ 11 算法更快，并且对所有k更快如果矩阵乘法指数为 2，则 ≥ 7。