AI 中文总结
研究图连通性的局部认证，开发利用连通性与组合结构联系的新方法，对\(k\)边连通性获\(O_k(\log n)\)位认证方案及下界，\(k\)顶点连通性在猜想下得\(\tilde{O}_k(\sqrt{n})\)位证书，还给出稀疏图类及一般图中\(k = 2\)时的相关结果。
AI 中文摘要
局部认证是一种仅使用局部信息验证全局图属性的框架。证明者为图的顶点分配称为证书的短标签，顶点与邻居交换证书并进行局部检查以确定图是否满足所需属性。此前研究给出了某些连通性的\(O(\log n)\)位证书。本文研究图连通性的局部认证，开发新方法，利用连通性与组合结构的联系。对于\(k\)边连通性，得到\(O_k(\log n)\)位认证方案及下界；对于\(k\)顶点连通性，在猜想下得到\(\tilde{O}_k(\sqrt{n})\)位证书。还表明在稀疏图类中\(k = 2\)时对数障碍可打破，一般图中\(2\)顶点连通性有\(\Omega(\log(\log^\ast n))\)位下界。
英文摘要
Local certification is a framework for verifying global graph properties using only local information. In this model, a prover assigns short labels, called certificates, to the vertices of a graph. Each vertex then exchanges certificates with its neighbors and performs a purely local check to determine whether the graph satisfies the desired property. This line of research has led to efficient certification schemes for a broad range of graph classes, including minor-closed families, topological graph classes, and graphs defined by forbidden subgraphs. In this paper, we study the local certification of graph connectivity. Prior work by Bousquet, Feuilloley, and Pierron (JPDC 2024) showed that $2$-vertex-connectivity, $2$-edge-connectivity, and $3$-vertex-connectivity admit $O(\log n)$-bit certificates, leveraging structural characterizations such as ear decompositions. We go substantially beyond these cases and investigate general $k$-vertex-connectivity and $k$-edge-connectivity. We develop new approaches that exploit connections between connectivity and combinatorial structures, including branchings, Eulerian subgraphs, and independent spanning trees. For $k$-edge-connectivity, we obtain an $O_k(\log n)$-bit certification scheme and prove a matching $Ω_k(\log n)$ lower bound for every $k\ge 3$. The lower bound also applies to $k$-vertex-connectivity. For $k$-vertex-connectivity, we obtain $\tilde{O}_k(\sqrt{n})$-bit certificates for every $k$ under a conjecture of Itai and Zehavi. We further show that, for $k=2$, the logarithmic barrier can be broken on sparse graph classes: $2$-edge-connectivity admits constant-size certificates in bounded-expansion graphs, and $2$-vertex-connectivity admits constant-size certificates in bounded-degree graphs. In contrast, for $2$-vertex-connectivity in general graphs, we prove an $Ω(\log(\log^\ast n))$-bit lower bound.
CommentsAbstract shortened to meet arXiv requirements
