We study the computational complexity of several problems connected with finding a maximal distance-$k$ matching of minimum cardinality or minimum weight in a given graph. We introduce the class of $k$-equimatchable graphs which is an edge analogue of $k$-equipackable graphs. We prove that the recognition of $k$-equimatchable graphs is co-NP-complete for any fixed $k \ge 2$. We provide a simple characterization for the class of strongly chordal graphs with equal $k$-packing and $k$-domination numbers. We also prove that for any fixed integer $\ell \ge 1$ the problem of finding a minimum weight maximal distance-$2\ell$ matching and the problem of finding a minimum weight $(2 \ell - 1)$-independent dominating set cannot be approximated in polynomial time in chordal graphs within a factor of $δ\ln |V(G)|$ unless $\mathrm{P} = \mathrm{NP}$, where $δ$ is a fixed constant (thereby improving the NP-hardness result of Chang for the independent domination case). Finally, we show the NP-hardness of the minimum maximal induced matching and independent dominating set problems in large-girth planar graphs.
Note: This version (as compared to the journal submission) contains corrections to Section 4.
Maximum matchings and minimum dominating sets in Apollonian networks and extended Tower of Hanoi graphs
阿波罗网络显示出显着的幂律和小世界性质，在大多数现实的网络系统中观察到的。它们的对偶图是推广的Tower of Hanoi图，它是由Tower of Hanoi图通过添加一个特殊的顶点连接到它的所有三个极端顶点而得到的。本文研究了Apollonian网络及其对偶图中的最大匹配和最小控制集，这两个问题在复杂网络的结构可控性等领域有着广泛的应用。对于这两个网络，我们确定它们的匹配数，控制数，最大匹配数，以及最小控制集的数量。
On a k-matching algorithm and finding k-factors in random graphs with minimum degree k+1 in linear time
证明了对$k+1\geq 3$和$c>(k+1)/2$ w. h. p.，关于$n$顶点、$cn$边和最小度$k+1$的随机图包含（近）完美$k$ -匹配.作为直接结果，我们得到了$G_{n,p}$的$(k+1)$ -核如果非空，则它跨越一个（近）生成$k$ -正则子图.这改进了Chan和Molloy的一个结果，并完全解决了Bollobás，Kim和Verstraëte的一个猜想.此外，我们证明了w. h. p.这样的子图可以在线性时间内找到。一个重要的元素的证明是一个随机算法的分析，找到$k$ -匹配的随机图与最小程度的$k+1$。
