A subset \(M\subseteq E\) of edges of a graph \(G=(V,E)\) is called a matching in G if no two edges in M share a common vertex. A matching M in G is called an induced matching if G[M], the subgraph of G induced by M, is the same as G[S], the subgraph of G induced by \(S=\{v \in V |\)v is incident on an edge of \(M\}\). The Maximum Induced Matching problem is to find an induced matching of maximum cardinality. Given a graph G and a positive integer k, the Induced Matching Decision problem is to decide whether G has an induced matching of cardinality at least k. The Maximum Weight Induced Matching problem in a weighted graph \(G=(V,E)\) in which the weight of each edge is a positive real number, is to find an induced matching such that the sum of the weights of its edges is maximum. It is known that the Induced Matching Decision problem and hence the Maximum Weight Induced Matching problem is known to be NP-complete for general graphs and bipartite graphs. In this paper, we strengthened this result by showing that the Induced Matching Decision problem is NP-complete for star-convex bipartite graphs, comb-convex bipartite graphs, and perfect elimination bipartite graphs, the subclasses of the class of bipartite graphs. On the positive side, we propose polynomial time algorithms for the Maximum Weight Induced Matching problem for circular-convex bipartite graphs and triad-convex bipartite graphs by making polynomial time reductions from the Maximum Weight Induced Matching problem in these graph classes to the Maximum Weight Induced Matching problem in convex bipartite graphs.

中文翻译：

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二重图某些子类中的最大权重诱导匹配

一个子集\（M \ subseteqË\）的曲线图的边的\（G =（V，E）\）被称为匹配在ģ如果在任何两个边缘中号共用共同的顶点。的匹配中号在ģ被称为导出匹配如果ģ [中号]，的子图G ^诱导中号，相同ģ [小号]的子图G ^诱导\（S = \ {V \在V | \）v入射在\（M \} \）的边缘上。该最大感应匹配问题是找到最大基数的诱导匹配。给定一个图G和一个正整数k，则诱导匹配决策问题是确定G是否具有至少为k的基数诱导匹配。加权图\（G =（V，E）\）中的最大权重诱导匹配问题（其中每个边的权重为正实数）是为了找到诱导匹配，使得其边的权重之和最大。众所周知，诱导匹配决策问题以及最大权重诱导匹配对于一般图和二部图，已知该问题是NP完全的。在本文中，我们通过证明星型凸二分图，梳状凸二分图和完全消除二分图是二分图类的子类的诱导匹配决策问题是NP完全的，从而加强了该结果。从积极的方面来说，我们通过将多项式时间从这些图类中的最大权重匹配问题减少到最大权重引起的多项式时间，提出了用于圆凸二分图和三重凸二分图的最大权重匹配问题的多项式时间算法。凸二部图中的匹配问题。