In 2005, Goddard, Hedetniemi, Hedetniemi and Laskar [Generalized subgraph-restricted matchings in graphs, Discrete Mathematics, 293 (2005) 129 - 138] asked the computational complexity of determining the maximum cardinality of a matching whose vertex set induces a disconnected graph. In this paper we answer this question. In fact, we consider the generalized problem of finding $c$-disconnected matchings; such matchings are ones whose vertex sets induce subgraphs with at least $c$ connected components. We show that, for every fixed $c \geq 2$, this problem is NP-complete even if we restrict the input to bounded diameter bipartite graphs, while can be solved in polynomial time if $c = 1$. For the case when $c$ is part of the input, we show that the problem is NP-complete for chordal graphs, while being solvable in polynomial time for interval graphs. Finally, we explore the parameterized complexity of the problem. We present an FPT algorithm under the treewidth parameterization, and an XP algorithm for graphs with a polynomial number of minimal separators when parameterized by $c$. We complement these results by showing that, unless NP $\subseteq$ coNP/poly, the related Induced Matching problem does not admit a polynomial kernel when parameterized by vertex cover and size of the matching nor when parameterized by vertex deletion distance to clique and size of the matching. As for Connected Matching, we show how to obtain a maximum connected matching in linear time given an arbitrary maximum matching in the input.
翻译:2005年,Godard, Hedetniemi, Hedetniemi, Hedetniemi 和 Laskar[图表中的普通子限制比对,Discrete Mathemats, 293(2005) 293(2005) 129 - 138] 询问确定匹配的最大基点的计算复杂性,以确定其顶点设置引发断开的图形。 在本文中,我们回答这个问题。 事实上, 我们考虑到寻找 $c 与该输入不相连接的匹配的普遍问题。 这种匹配是那些顶端设置至少能带来美元连接组件的子节点。 我们显示,对于每个固定的 $c\ geq 2 中每个固定的子节节点匹配,即使我们把输入限制在直线性双部图中, 293 129 - 129 129 - 138 问确定匹配的最大基本基点的基点。 而对于当美元是输入的一部分时, 当我们通过多调调时, 我们的基点的基点的基点可以连接到如何连接。 最后,我们通过直径直线 直径的基点的比值的基点的基点的基点 显示一个直线值的比值的基点的基点的基点的基点的比值的复杂性, 我们的基点的比值的比值的比值的比值的比值的比值, 。