关联解离、独立和匹配
Relating dissociation, independence, and matchings
摘要：
图中的解离集是诱导最大度至多为$1$的子图的顶点的集合。计算给定图$G$的解离数${\rm diss}(G)$（定义为$G$中最大解离集的阶数）在算法上是困难的，即使当$G$被限制为二分时也是如此。最近，Hosseinian和Butenko提出了一个简单的$\frac{4}{3}$ -近似算法的二部图的解离数问题。他们的结果依赖于他们工作中隐含的不等式${\rm diss}(G)\leq\frac{4}{3}α(G-M)$，其中$G$是二分图，$M$是$G$中的最大匹配，$α(G-M)$表示$G-M$的独立数。我们表明，对$(G,M)$，这个不等式成立的平等，可以有效地识别，并且可以有效地确定它们的最大解离集。图$G$的解离数满足$\max\{ α(G),2ν_s(G)\} \leq {\rm diss}(G)\leq α(G)+ν_s(G)\leq 2α(G)$，其中$ν_s(G)$表示$G$的诱导匹配数。我们发现，决定是否${\rm diss}(G)$等于任何四个术语的下限和上限${\rm diss}(G)$是NP-困难的。
英文摘要：
A dissociation set in a graph is a set of vertices inducing a subgraph of maximum degree at most $1$. Computing the dissociation number ${\rm diss}(G)$ of a given graph $G$, defined as the order of a maximum dissociation set in $G$, is algorithmically hard even when $G$ is restricted to be bipartite. Recently, Hosseinian and Butenko proposed a simple $\frac{4}{3}$-approximation algorithm for the dissociation number problem in bipartite graphs. Their result relies on the inequality ${\rm diss}(G)\leq\frac{4}{3}α(G-M)$ implicit in their work, where $G$ is a bipartite graph, $M$ is a maximum matching in $G$, and $α(G-M)$ denotes the independence number of $G-M$. We show that the pairs $(G,M)$ for which this inequality holds with equality can be recognized efficiently, and that a maximum dissociation set can be determined for them efficiently. The dissociation number of a graph $G$ satisfies $\max\{ α(G),2ν_s(G)\} \leq {\rm diss}(G)\leq α(G)+ν_s(G)\leq 2α(G)$, where $ν_s(G)$ denotes the induced matching number of $G$. We show that deciding whether ${\rm diss}(G)$ equals any of the four terms lower and upper bounding ${\rm diss}(G)$ is NP-hard.
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