1,134 research outputs found
On some special classes of contact -VPG graphs
A graph is a -VPG graph if one can associate a path on a rectangular
grid with each vertex such that two vertices are adjacent if and only if the
corresponding paths intersect at at least one grid-point. A graph is a
contact -VPG graph if it is a -VPG graph admitting a representation
with no two paths crossing and no two paths sharing an edge of the grid. In
this paper, we present a minimal forbidden induced subgraph characterisation of
contact -VPG graphs within four special graph classes: chordal graphs,
tree-cographs, -tidy graphs and -free graphs. Moreover, we present a
polynomial-time algorithm for recognising chordal contact -VPG graphs.Comment: 34 pages, 15 figure
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
Characterizing 2-crossing-critical graphs
It is very well-known that there are precisely two minimal non-planar graphs:
and (degree 2 vertices being irrelevant in this context). In
the language of crossing numbers, these are the only 1-crossing-critical
graphs: they each have crossing number at least one, and every proper subgraph
has crossing number less than one. In 1987, Kochol exhibited an infinite family
of 3-connected, simple 2-crossing-critical graphs. In this work, we: (i)
determine all the 3-connected 2-crossing-critical graphs that contain a
subdivision of the M\"obius Ladder ; (ii) show how to obtain all the
not 3-connected 2-crossing-critical graphs from the 3-connected ones; (iii)
show that there are only finitely many 3-connected 2-crossing-critical graphs
not containing a subdivision of ; and (iv) determine all the
3-connected 2-crossing-critical graphs that do not contain a subdivision of
.Comment: 176 pages, 28 figure
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