Linear‐time algorithms for eliminating claws in graphs

Abstract Since many ‐complete graph problems are polynomial‐time solvable when restricted to claw‐free graphs, we study the problem of determining the distance of a given graph to a claw‐free graph, considering vertex elimination a measure. Claw‐free Vertex Deletion (CFVD) consists of determining th...

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Bibliographic Details
Published in:International transactions in operational research 2024-01, Vol.31 (1), p.296-315
Main Authors: Bonomo‐Braberman, Flavia, Nascimento, Julliano R., Oliveira, Fabiano S., Souza, Uéverton S., Szwarcfiter, Jayme L.
Format: Article
Language:English
Subjects:
Algorithms
Apexes
Graph theory
Graphs
Mathematical analysis
Operations research
Polynomials
Trees (mathematics)
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Summary:Abstract Since many ‐complete graph problems are polynomial‐time solvable when restricted to claw‐free graphs, we study the problem of determining the distance of a given graph to a claw‐free graph, considering vertex elimination a measure. Claw‐free Vertex Deletion (CFVD) consists of determining the minimum number of vertices to be removed from a graph such that the resulting graph is claw‐free. Although CFVD is ‐hard in general and recognizing claw‐free graphs is still a challenge, where the current best deterministic algorithm for a graph G consists of performing executions of the best algorithm for matrix multiplication, we present linear‐time algorithms for CFVD on weighted block graphs and weighted graphs with bounded treewidth. Furthermore, we show that this problem on forests can be solved in linear time by a simpler algorithm, and we determine the exact values for full k ‐ary trees. On the other hand, we show that CFVD is ‐hard even when the input graph is a split graph. We also show that the problem is hard to be approximated within any constant factor better than 2, assuming the unique games conjecture.
ISSN:0969-6016
1475-3995
DOI:10.1111/itor.13100