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The k-in-a-Path Problem for Claw-free Graphs
2010
Algorithmica
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We show how to solve this problem in polynomial time on claw-free graphs, when k is not part of the input but an arbitrarily fixed integer. ...
Testing whether there is an induced path in a graph spanning k given vertices is already NP-complete in general graphs when k=3. ...
Acknowledgments We would like to thank the two anonymous referees for their helpful comments. ...
doi:10.1007/s00453-010-9468-z
fatcat:76mfyeqqvrfq3jo2ucurnkggmq
THE k-IN-A-PATH PROBLEM FOR CLAW-FREE GRAPHS
2010
Symposium on Theoretical Aspects of Computer Science
unpublished
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We show how to solve this problem in polynomial time on claw-free graphs, when k is not part of the input but an arbitrarily fixed integer. ...
Testing whether there is an induced path in a graph spanning k given vertices is already NP-complete in general graphs when k = 3. ...
Notice that any induced tree in a claw-free graph is in fact an induced path. Observation 1.2. The k-in-a-Path and k-in-a-Tree problem are equivalent for the class of claw-free graphs. Motivation. ...
fatcat:6cjay7p3pbba7ag4haintyhfye
Induced Disjoint Paths in Claw-Free Graphs
[chapter]
2012
Lecture Notes in Computer Science
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Several related problems, such as the k-in-a-Path problem, are proved to be fixed-parameter tractable for claw-free graphs as well. ...
We show that this problem is fixed-parameter tractable for claw-free graphs when parameterized by k. ...
We would like to thank two anonymous reviewers for their comments, which helped us improve the readability of our paper. ...
doi:10.1007/978-3-642-33090-2_45
fatcat:dskh3ozjpbh5lmjcptcgjb6vte
Induced Disjoint Paths in Claw-Free Graphs
[article]
2014
arXiv
pre-print
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Several related problems, such as the k-in-a-Path problem, are proven to be fixed-parameter tractable for claw-free graphs as well. ...
We show that this problem is fixed-parameter tractable for claw-free graphs when parameterized by k. ...
The problems k-in-a-Cycle, k-in-a-Path, and k-in-a-Tree are fixed-parameter tractable for claw-free graphs when parameterized by k. Proof. First we consider the k-in-a-Cycle problem. ...
arXiv:1202.4419v2
fatcat:gukgczgypzcttdbwqn4bxqswnm
Induced Disjoint Paths in Claw-Free Graphs
2015
SIAM Journal on Discrete Mathematics
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Several related problems, such as the k-in-a-Path problem, are proved to be fixed-parameter tractable for claw-free graphs as well. ...
We show that this problem is fixed-parameter tractable for claw-free graphs when parameterized by k. ...
We would like to thank two anonymous reviewers for their comments, which helped us improve the readability of our paper. ...
doi:10.1137/140963200
fatcat:t4rpbc6wgjfercr7dr6ogilzli
Packing 3-vertex paths in claw-free graphs and related topics
2011
Discrete Applied Mathematics
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Let G be a 3-connected claw-free graph, x a vertex in G, e = xy an edge in G, and P a 3-vertex path in G. ...
A claw is a graph with four vertices and three edges incident to the same vertex. A graph is claw-free if it has no induced subgraph isomorphic to a claw. Our results include the following. ...
(c12) We show that the -packing problem for a claw-free graph G can be reduced in polynomial time to that for a special claw-free graph K (called a cactus) with v(K ) ≤ v(G) (see 4.5 and 4.6). ...
doi:10.1016/j.dam.2010.05.001
fatcat:74tl7ztnxrdt3ccymbrplkpbay
Deletion to Scattered Graph Classes II – Improved FPT Algorithms for Deletion to Pairs of Graph Classes
[article]
2022
arXiv
pre-print
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The problem of deletion to Π, takes as input a graph G and asks for a minimum number (or a fixed integer k) of vertices to be deleted from G so that the resulting graph belongs to Π. ...
Recently, the study of a natural extension of the problem was initiated where we are given a finite set of hereditary graph classes, and the goal is to determine whether k vertices can be deleted from ...
Claw-free or Triangle-Free graphs We define the problem.
Claw-Free-or-Triangle-Free Deletion Input: An undirected graph G = (V, E) and an integer k. ...
arXiv:2201.03142v1
fatcat:h6f66ushvnft3ng3aebxx2sayy
Reconfiguring Independent Sets in Claw-Free Graphs
[chapter]
2014
Lecture Notes in Computer Science
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We present a polynomial-time algorithm that, given two independent sets in a claw-free graph G, decides whether one can be transformed into the other by a sequence of elementary steps. ...
Each elementary step is to remove a vertex v from the current independent set S and to add a new vertex w (not in S) such that the result is again an independent set. ...
This motivated the question asked in [11] whether for connected, claw-free and evenhole-free graph G, TS k (G) is connected. ...
doi:10.1007/978-3-319-08404-6_8
fatcat:ktyhlpltubeefktjedu6b7kon4
Finding Induced Paths of Given Parity in Claw-Free Graphs
2010
Algorithmica
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Our results imply that the problem of deciding if two specified vertices of a claw-free graph form an even pair, as well as the problem of deciding if a given claw-free graph has an even pair, can be solved ...
Although all three problems are NP-complete in general, we show that they can be solved in O(n 5 ) time for the class of claw-free graphs. ...
The authors would like to thank Jiří Fiala, Benjamin Lévêque, Luke Mathieson, Nicolas Trotignon and two anonymous reviewers for helpful suggestions. ...
doi:10.1007/s00453-010-9470-5
fatcat:emwqsmiedbfrnc4qd35kei5xki
Finding Induced Paths of Given Parity in Claw-Free Graphs
[chapter]
2010
Lecture Notes in Computer Science
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Our results imply that the problem of deciding if two specified vertices of a claw-free graph form an even pair, as well as the problem of deciding if a given claw-free graph has an even pair, can be solved ...
Although all three problems are NP-complete in general, we show that they can be solved in O(n 5 ) time for the class of claw-free graphs. ...
The authors would like to thank Jiří Fiala, Benjamin Lévêque, Luke Mathieson, Nicolas Trotignon and two anonymous reviewers for helpful suggestions. ...
doi:10.1007/978-3-642-11409-0_30
fatcat:hstidpzhcjgvzka2vl6sd6yxma
On linear and circular structure of (claw, net)-free graphs
2003
Discrete Applied Mathematics
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We show also how one can use structural properties of (claw, net)-free graphs to solve e ciently the domination, independent domination, and independent set problems on these graphs. ? ...
Moreover, using LexBFS we present a linear time algorithm which, for a given (claw, net)-free graph, ÿnds either a dominating pair or an induced doubly dominating cycle. ...
In [22] , Hempel and Kratsch gave linear time algorithms for all three problems on the class of claw-free AT-free graphs. ...
doi:10.1016/s0166-218x(02)00571-1
fatcat:2pbmjgfx7zfj7hwy3cz2djmtlm
Colouring Generalized Claw-Free Graphs and Graphs of Large Girth: Bounding the Diameter
[article]
2021
arXiv
pre-print
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A graph G is H-free if G does not contain H as an induced subgraph. It is known that for all k≥ 3, the k-Colouring problem is NP-complete for H-free graphs if H contains an induced claw or cycle. ...
For a fixed integer, the k-Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for an integer k, such that no two adjacent vertices are coloured alike. ...
What are the complexities of 3-Colouring for K 1 1,4 -free graphs of diameter 3 and for K 2 1,3 -free graphs of diameter 3? Open Problem 4. ...
arXiv:2111.11897v1
fatcat:w76s7sr3oze47fuxugd5767g4i
Intersection Graphs of Non-crossing Paths
[article]
2020
arXiv
pre-print
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A direct consequence of our certifying algorithms is a linear time algorithm certifying the presence/absence of an induced claw (K_1,3) in a chordal graph. ...
For the intersection graphs of NC paths of a tree, we characterize the minimum connected dominating sets (leading to a linear time algorithm to compute one). ...
This is relevant for us because, as we will see in Section 3, the claw-free chordal graphs are precisely the NC-path-tree graphs. ...
arXiv:1907.00272v2
fatcat:paeipvyuk5gmxl6wcunfatj56a
On the Erdős-Gyárfás conjecture in claw-free graphs
[article]
2013
arXiv
pre-print
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In this paper, we obtain some results on this question, in particular for cubic claw-free graphs. ...
Since this conjecture has proven to be far from reach, Hobbs asked if the Erdős-Gyárfás conjecture holds in claw-free graphs. ...
In the last section, we discuss on the conjecture in cubic claw-free graphs. We also pose a related conjecture and a problem. ...
arXiv:1109.5398v3
fatcat:u5vut7kttbar7pakkg7zktj7bu
From matchings to independent sets
2017
Discrete Applied Mathematics
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It is well-known that finding a maximum matching in G is equivalent to finding a maximum independent set in the line graph of G. For general graphs, the maximum independent set problem is NP-hard. ...
We also review various techniques that may lead to efficient algorithms for the maximum independent set problem in restricted graph families, with a focus given to augmenting graphs and graph transformations ...
For the same reason, paths with an even number of vertices cannot be augmenting. Therefore, every connected claw-free augmenting graph is a path P k with odd k. ...
doi:10.1016/j.dam.2016.04.012
fatcat:vf4xsf7is5fxni5ullpnngptqq
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