• Corpus ID: 239009976

A Note on Coloring $(4K_1, C_4, C_6)$-free graphs with a $C_7$

@inproceedings{Koutecky2021ANO,
  title={A Note on Coloring \$(4K\_1, C\_4, C\_6)\$-free graphs with a \$C\_7\$},
  author={Martin Kouteck'y},
  year={2021}
}
Even-hole-free graphs are a graph class of much interest. Foley et al. [Graphs Comb. 36(1): 125-138 (2020)] have recently studied (4K1, C4, C6)-free graphs, which form a subclass of even-holefree graphs. Specifically, Foley et al. have shown an algorithm for coloring these graphs via bounded clique-width if they contain a C7. In this note, we give a simpler and much faster algorithm via a more restrictive graph parameter, neighborhood diversity. 

References

SHOWING 1-10 OF 13 REFERENCES
The Intersection of Two Vertex Coloring Problems
TLDR
This paper presents partial results on the problem of coloring even hole-free graphs and describes the construction of a graph G that is L -free if G does not contain any graph in L as an induced subgraph.
Cliquewidth III: The Odd Case of Graph Coloring Parameterized by Cliquewidth
TLDR
GC is the first natural problem known to require exponential dependence on the parameter in the exponent of n and is closed by proving a lower bound of n2o(k), which shows that GC behaves qualitatively different from the other three problems.
Parameterized Algorithms for Modular-Width
TLDR
It is argued that modular-width occupies a sweet spot as a graph parameter, generalizing several simpler notions on dense graphs but still evading the "price of generality" paid by clique-width.
A faster algorithm to recognize even-hole-free graphs
TLDR
The problem of determining whether an n-node m-edge graph has an even hole, i.e., an induced simple cycle consisting of an even number of nodes, is solved in time O(n11).
Decomposition of even-hole-free graphs with star cutsets and 2-joins
Algorithmic Meta-theorems for Restrictions of Treewidth
  • M. Lampis
  • Mathematics, Computer Science
    Algorithmica
  • 2011
TLDR
It is shown that any FO property in both of these classes with a singly exponential parameter dependence is possible and that it is possible to decide MSO logic on graphs of bounded vertex cover with a doubly exponentialParameter dependence, and is proved that the upper bound results cannot be improved significantly, under widely believed complexity assumptions.
Even-hole-free graphs part II: Recognition algorithm
TLDR
An algorithm that determines in polytime whether a graph contains an even hole is presented, based on a decomposition theorem for even-hole-free graphs obtained in Part I of this paper.
Even-hole-free graphs part I: Decomposition theorem
TLDR
A decomposition theorem for even-hole-free graphs is proved and this theorem is used in the second part of this paper to obtain a polytime recognition algorithm for even -hole- free graphs.
Evaluating and Tuning n-fold Integer Programming
TLDR
The original algorithm is practically unusable, but a series of improvements are discovered which make its evaluation possible and a new strategy for finding "approximatelly best" steps wildly outperforms the original construction.
An Algorithmic Theory of Integer Programming
TLDR
It is shown that integer programming can be solved in time, and a strongly-polynomial algorithm is derived, that is, with running time $g(a,d)\textrm{poly}(n)$, independent of the rest of the input data.
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