60 research outputs found
FPT algorithms to recognize well covered graphs
Given a graph , let and be the sizes of a minimum and a
maximum minimal vertex covers of , respectively. We say that is well
covered if (that is, all minimal vertex covers have the same
size). Determining if a graph is well covered is a coNP-complete problem. In
this paper, we obtain -time and -time
algorithms to decide well coveredness, improving results of Boria et. al.
(2015). Moreover, using crown decomposition, we show that such problems admit
kernels having linear number of vertices. In 2018, Alves et. al. (2018) proved
that recognizing well covered graphs is coW[2]-hard when the independence
number is the parameter. Contrasting with such
coW[2]-hardness, we present an FPT algorithm to decide well coveredness when
and the degeneracy of the input graph are aggregate parameters.
Finally, we use the primeval decomposition technique to obtain a linear time
algorithm for extended -laden graphs and -graphs, which is FPT
parameterized by , improving results of Klein et al (2013).Comment: 15 pages, 2 figure
Minimal classes of graphs of unbounded clique-width defined by finitely many forbidden induced subgraphs
We discover new hereditary classes of graphs that are minimal (with respect
to set inclusion) of unbounded clique-width. The new examples include split
permutation graphs and bichain graphs. Each of these classes is characterised
by a finite list of minimal forbidden induced subgraphs. These, therefore,
disprove a conjecture due to Daligault, Rao and Thomasse from 2010 claiming
that all such minimal classes must be defined by infinitely many forbidden
induced subgraphs.
In the same paper, Daligault, Rao and Thomasse make another conjecture that
every hereditary class of unbounded clique-width must contain a labelled
infinite antichain. We show that the two example classes we consider here
satisfy this conjecture. Indeed, they each contain a canonical labelled
infinite antichain, which leads us to propose a stronger conjecture: that every
hereditary class of graphs that is minimal of unbounded clique-width contains a
canonical labelled infinite antichain.Comment: 17 pages, 7 figure
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