Linear-time Algorithms for Eliminating Claws in Graphs

Flavia Bonomo-Braberman; Julliano R. Nascimento; Fabiano S. Oliveira; Uéverton S. Souza; Jayme L. Szwarcfiter

doi:10.1111/itor.13100

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arXiv · 2004.05672

Linear-time Algorithms for Eliminating Claws in Graphs

Flavia Bonomo-Braberman·Julliano R. Nascimento·Fabiano S. Oliveira·Uéverton S. Souza·Jayme L. Szwarcfiter

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Abstract

Since many NP-complete graph problems have been shown 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 as 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 NP-complete in general and recognizing claw-free graphs is still a challenge, where the current best algorithm for a graph $G$ has the same running time 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 can be solved in linear time by a simpler algorithm on forests, and we determine the exact values for full $k$-ary trees. On the other hand, we show that CLAW-FREE VERTEX DELETION is NP-complete even when the input graph is a split graph. We also show that the problem is hard to approximate within any constant factor better than $2$, assuming the Unique Games Conjecture.

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Flavia Bonomo-Braberman, Julliano R. Nascimento, Fabiano S. Oliveira, Uéverton S. Souza, Jayme L. Szwarcfiter. 2020-04-12. Linear-time Algorithms for Eliminating Claws in Graphs. https://doi.org/10.1111/itor.13100

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