Given a graph $G=(V,E)$, for a vertex set $S\subseteq V$, let $N(S)$ denote the set of vertices in $V$ that have a neighbor in $S$. Extending the concept of binding number of graphs by Woodall~(1973), for a vertex set $X \subseteq V$, we define the binding number of $X$, denoted by $\bind(X)$, as the maximum number $b$ such that for every $S \subseteq X$ where $N(S)\neq V(G)$ it holds that $|N(S)|\ge b {|S|}$. Given this definition, we prove that if a graph $V(G)$ contains a subset $X$ with $\bind(X)= 1/k$ where $k$ is an integer, then $G$ possesses a matching of size at least $|X|/(k+1)$. Using this statement, we derive tight bounds for the estimators of the matching size in planar graphs. These estimators are previously used in designing sublinear space algorithms for approximating the maching size in the data stream model of computation. In particular, we show that the number of locally superior vertices is a $3$ factor approximation of the matching size in planar graphs. The previous analysis by Jowhari (2023) proved a $3.5$ approximation factor. As another application, we show a simple variant of an estimator by Esfandiari \etal (2015) achieves $3$ factor approximation of the matching size in planar graphs. Namely, let $s$ be the number of edges with both endpoints having degree at most $2$ and let $h$ be the number of vertices with degree at least $3$. We prove that when the graph is planar, the size of matching is at least $(s+h)/3$. This result generalizes a known fact that every planar graph on $n$ vertices with minimum degree $3$ has a matching of size at least $n/3$.

翻译：给定图$G=(V,E)$，对于顶点子集$S\subseteq V$，令$N(S)$表示$V$中与$S$存在邻接关系的顶点集合。基于Woodall (1973)提出的图绑定数概念，对顶点子集$X \subseteq V$，我们定义$X$的绑定数$\bind(X)$为满足以下条件的最大数$b$：对任意满足$N(S)\neq V(G)$的$S \subseteq X$，均有$|N(S)|\ge b |S|$。基于这一定义，我们证明：若图$V(G)$包含子集$X$且$\bind(X)= 1/k$（其中$k$为整数），则$G$存在大小至少为$|X|/(k+1)$的匹配。利用该结论，我们推导了平面图中匹配规模估计量的紧界。这些估计量此前被用于设计数据流计算模型中近似匹配规模的次线性空间算法。特别地，我们证明：在平面图中，"局部优顶点"数量是匹配规模的3因子近似，而Jowhari (2023)的先前分析证明了3.5近似因子。另一应用体现在：我们展示了Esfandiari等(2015)提出的估计量的一个简单变体，可在平面图中实现匹配规模的3因子近似。具体而言，令$s$为两个端点度数均不超过2的边数，$h$为度数至少为3的顶点数。我们证明：当图为平面图时，匹配规模至少为$(s+h)/3$。该结论推广了一个已知事实——每个具有最小度3的$n$顶点平面图均存在大小至少为$n/3$的匹配。