In [4] it was shown that in a 5‐connected even planar triangulation G, every matching M of size $|M|<|V(G)|/2$ can be extended to a perfect matching of G, as long as the edges of M lie at distance at least 5 from each other. Somewhat later in [7], the following result was proved. Let $G$ be a 5‐connected triangulation of a surface $\mathrm{\Sigma }$ different from the sphere. Let $\chi =\chi \left(\mathrm{\Sigma }\right)$ be the Euler characteristic of $\mathrm{\Sigma }$. Suppose $V_0\subset V\left(G\right)$ with $|V\left(G\right)-V_0|$ even and $M$ and $N$ are two matchings in $G-V_0$ such that $M\cap N=\varnothing$. Further suppose that the pairwise distance between two elements of $V_0\cup M\cup N$ is at least 5 and the face‐width of the embedding of $G$ in $\mathrm{\Sigma }$ is at least $\max \left\{20|M|-8\chi -23,6\right\}$. Then there is a perfect matching $M_0$ in $G-V_0$ which contains $M$ such that $M_0\cap N=\varnothing$. In the present paper, we present some results which, in a sense, lie in the gap between the two above theorems, in that they deal with restricted matching extension in a planar triangulation when a set of vertices which lie pairwise at sufficient distance from one another has been deleted. In particular, we prove a planar analogue of the result in [7] stated above.

中文翻译：

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距离受限的匹配扩展缺少平面的5个连接的三角剖分中的顶点和边

在[4]中表明，在5个连接的偶数平面三角剖分G中，每个匹配的M的大小$|\mathrm{中号}|<|V（G）|/2$可以扩展为G的完美匹配，只要M的边缘彼此之间的距离至少为5。在[7]中的稍后部分，证明了以下结果。让$G$ 是表面的5连接三角剖分 $\mathrm{\Sigma }$ 与球体不同。让$\chi =\chi （\mathrm{\Sigma }）$ 是欧拉的特征 $\mathrm{\Sigma }$。假设$V_0\subset V（G）$ 与 $|V（G）-V_0|$ 甚至和 $\mathrm{中号}$ 和 $ñ$ 是两个匹配项 $G-V_0$ 这样 $\mathrm{中号}\cap ñ=\varnothing$。进一步假设的两个元素之间的成对距离$V_0\cup \mathrm{中号}\cup ñ$ 至少为5，且嵌入的面宽度 $G$ 在 $\mathrm{\Sigma }$ 至少是 $\mathrm{最高}\left\{20|\mathrm{中号}|-8\chi -23，6\right\}$。然后有一个完美的匹配${\mathrm{中号}}_0$ 在 $G-V_0$ 其中包含 $\mathrm{中号}$ 这样 ${\mathrm{中号}}_0\cap ñ=\varnothing$。在本文中，我们提出了一些结果，从某种意义上说，它们位于上述两个定理之间的缝隙中，因为当一组顶点成对放置且距离一个顶点足够远时，它们处理平面三角剖分中的受限匹配扩展另一个已被删除。特别是，我们证明了上述[7]中结果的平面类似物。