Augustine et al. [DISC 2022] initiated the study of distributed graph algorithms in the presence of Byzantine nodes in the congested clique model. In this model, there is a set $B$ of Byzantine nodes, where $|B|$ is less than a third of the total number of nodes. These nodes have complete knowledge of the network and the state of other nodes, and they conspire to alter the output of the system. The authors addressed the connectivity problem, showing that it is solvable under the promise that either the subgraph induced by the honest nodes is connected, or the graph has $2|B|+1$ connected components. In the current work, we continue the study of the Byzantine congested clique model by considering the recognition of other graph properties, specifically hereditary properties. A graph property is hereditary if it is closed under taking induced subgraphs. Examples of hereditary properties include acyclicity, bipartiteness, planarity, and bounded (chromatic, independence) number, etc. For each class of graphs ${\bf G}$ satisfying a hereditary property (a hereditary graph-class), we propose a randomized algorithm which, with high probability, (1) accepts if the input graph $G$ belongs to ${\bf G}$, and (2) rejects if $G$ contains at least $|B| + 1$ disjoint subgraphs not belonging to ${\bf G}$. The round complexity of our algorithm is $$O\left(\left(\dfrac{\log \left(\left|{\bf G}_n\right|\right)}{n} +|B|\right)\cdot\textrm{polylog}(n)\right),$$ where ${\bf G}_n$ is the set of $n$-node graphs in ${\bf G}$. Finally, we obtain an impossibility result that proves that our result is tight. Indeed, we consider the hereditary class of acyclic graphs, and we prove that there is no algorithm that can distinguish between a graph being acyclic and a graph having $|B|$ disjoint cycles.

翻译：Augustine等人[DISC 2022]首次在拥塞团模型中研究了存在拜占庭节点的分布式图算法。在该模型中，存在一个拜占庭节点集合$B$，其中$|B|$小于总节点数的三分之一。这些节点完全了解网络拓扑及其他节点的状态，并合谋篡改系统输出。作者解决了连通性问题，证明在承诺诚实节点诱导的子图连通或该图具有$2|B|+1$个连通分量的前提下该问题可解。本文延续对拜占庭拥塞团模型的研究，重点关注其他图性质的识别问题，特别是遗传性质。若图性质在取诱导子图操作下封闭，则称为遗传性质。遗传性质的实例包括无环性、二部性、平面性及有界（色数、独立集）数等。对于每个满足遗传性质的图类${\\bf G}$（遗传图类），我们提出一个随机算法，该算法以高概率满足：(1)当输入图$G$属于${\\bf G}$时接受；(2)当$G$包含至少$|B|+1$个不属于${\\bf G}$的不相交子图时拒绝。算法轮复杂度为$$O\\left(\\left(\\dfrac{\\log \\left(\\left|{\\bf G}_n\\right|\\right)}{n} +|B|\\right)\\cdot\\textrm{polylog}(n)\\right),$$其中${\\bf G}_n$表示${\\bf G}$中具有$n$个节点的图集合。最后，我们通过不可能性结果证明本研究的紧确性：针对无环图的遗传类，我们证明不存在任何算法能区分一个图是否无环与一个图是否包含$|B|$个不相交环。