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引用次数: 0
摘要
给定一个属性(图类)Π、一个图 G 和一个整数 k,Π-补全问题包括判断我们是否能通过在 G 上添加最多 k 条边将 G 变成一个具有属性 Π 的图。众所周知,当 Π 是适当区间图 (PIG) 的属性时,Π-补全问题对于一般图来说是 NP-困难的。在这项工作中,我们研究了弦图不同子类中的 PIG-补全问题。我们证明,即使仅限于分裂图,该问题仍然是 NP-完全的。然后,我们将注意力转向正面结果,并提出了多项式时间算法,用于解决输入仅限于毛毛虫图和阈值图时的 PIG-补全问题。我们还提出了准阈值图的最小共边完成的高效算法,为该图类中的 PIG 完成问题提供了一个下界。
On the proper interval completion problem within some chordal subclasses
Given a property (graph class) Π, a graph G, and an integer k, the Π-completion problem consists of deciding whether we can turn G into a graph with the property Π by adding at most k edges to G. The Π-completion problem is known to be NP-hard for general graphs when Π is the property of being a proper interval graph (PIG). In this work, we study the PIG-completion problem within different subclasses of chordal graphs. We show that the problem remains NP-complete even when restricted to split graphs. We then turn our attention to positive results and present polynomial time algorithms to solve the PIG-completion problem when the input is restricted to caterpillar and threshold graphs. We also present an efficient algorithm for the minimum co-bipartite-completion for quasi-threshold graphs, which provides a lower bound for the PIG-completion problem within this graph class.
期刊介绍:
Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory.
Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.