The family of $(k,\ell)$-sparse graphs, introduced by Lorea, plays a central role in combinatorial optimization and has a wide range of applications, particularly in rigidity theory. A key algorithmic problem is to decide whether a given graph is $(k,\ell)$-sparse and, if not, to produce a vertex set certifying the failure of sparsity. While pebble game algorithms have long yielded $O(n^2)$-time recognition throughout the classical range $0 \leq \ell < 2k$, and $O(n^3)$-time algorithms in the extended range $2k \leq \ell < 3k$, substantially faster bounds were previously known only in a few special cases. We present new recognition algorithms for the parameter ranges $0 \le \ell \le k$, $k < \ell < 2k$, and $2k \leq \ell < 3k$. Our approach combines bounded-indegree orientations, reductions to rooted arc-connectivity, augmenting-path techniques, and a divide-and-conquer method based on centroid decomposition. This yields the first subquadratic, and in fact near-linear-time, recognition algorithms throughout the classical range when instantiated with the fastest currently available subroutines. Under purely combinatorial implementations, the running times become $O(n\sqrt n)$ for $0 \leq \ell \leq k$ and $O(n\sqrt{n\log n})$ for $k< \ell <2k$. For $2k \leq \ell < 3k$, we obtain an $O(n^2)$-time algorithm when $\ell \leq 2k+1$ and an $O(n^2\log n)$-time algorithm otherwise. In each case, the algorithm can also return an explicit violating set certifying that the input graph is not $(k,\ell)$-sparse.

翻译：由Lorea引入的$(k,\ell)$-稀疏图族在组合优化中扮演核心角色，并广泛应用于刚体理论等领域。关键算法问题在于判定给定图是否为$(k,\ell)$-稀疏图，若非如此，则需输出一个验证稀疏性失效的顶点集。经典参数范围$0 \leq \ell < 2k$内，卵石游戏算法早已实现$O(n^2)$时间的识别，而在扩展范围$2k \leq \ell < 3k$内则需$O(n^3)$时间算法；先前仅在少数特例中已知更快的界限。我们针对参数范围$0 \le \ell \le k$、$k < \ell < 2k$和$2k \leq \ell < 3k$分别提出新识别算法。该方法结合了有界入度定向、归约至根弧连通性、增广路径技术，以及基于质心分解的分治策略。当采用当前最快可用子程序时，这首次在经典参数范围内实现了亚二次时间、事实上是近线性时间的识别算法。若采用纯组合实现，则在$0 \leq \ell \leq k$范围内运行时间为$O(n\sqrt n)$，在$k< \ell <2k$范围内为$O(n\sqrt{n\log n})$。对于$2k \leq \ell < 3k$范围，当$\ell \leq 2k+1$时我们获得$O(n^2)$时间算法，其余情况则为$O(n^2\log n)$时间算法。每种情况下，算法均可返回显式违规集，以验证输入图并非$(k,\ell)$-稀疏图。