Disconnected Matchings
Guilherme C. M. Gomes, Bruno P. Masquio, Paulo E. D. Pinto, Vinicius, F. dos Santos, Jayme L. Szwarcfiter

TL;DR
This paper investigates the computational complexity of finding maximum matchings with a specified number of connected components, providing complexity results, algorithms, and parameterized analysis for various graph classes.
Contribution
It establishes NP-completeness results for c-disconnected matchings, offers polynomial algorithms for specific cases, and explores fixed-parameter tractability and kernelization bounds.
Findings
NP-complete for fixed c ≥ 2 on bounded diameter bipartite graphs
Polynomial-time solvable for c=1 and for interval graphs when c is part of input
FPT algorithm under treewidth and XP algorithm for graphs with polynomial minimal separators
Abstract
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 -disconnected matchings; such matchings are ones whose vertex sets induce subgraphs with at least connected components. We show that, for every fixed , this problem is NP-complete even if we restrict the input to bounded diameter bipartite graphs, while can be solved in polynomial time if . For the case when 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…
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Taxonomy
TopicsAdvanced Graph Theory Research · Complexity and Algorithms in Graphs · Optimization and Search Problems
