61 research outputs found
A survey of parameterized algorithms and the complexity of edge modification
The survey is a comprehensive overview of the developing area of parameterized algorithms for graph modification problems. It describes state of the art in kernelization, subexponential algorithms, and parameterized complexity of graph modification. The main focus is on edge modification problems, where the task is to change some adjacencies in a graph to satisfy some required properties. To facilitate further research, we list many open problems in the area.publishedVersio
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
Further topics in connectivity
Continuing the study of connectivity, initiated in §4.1 of the Handbook, we survey here some (sufficient) conditions under which a graph or digraph has a given connectivity or edge-connectivity. First, we describe results concerning maximal (vertex- or edge-) connectivity. Next, we deal with conditions for having (usually lower) bounds for the connectivity parameters. Finally, some other general connectivity measures, such as one instance of the so-called “conditional connectivity,” are considered.
For unexplained terminology concerning connectivity, see §4.1.Peer ReviewedPostprint (published version
Hardness Transitions of Star Colouring and Restricted Star Colouring
We study how the complexity of the graph colouring problems star colouring
and restricted star colouring vary with the maximum degree of the graph.
Restricted star colouring (in short, rs colouring) is a variant of star
colouring. For , a -colouring of a graph is a function
such that for every edge of
. A -colouring of is called a -star colouring of if there is
no path in with and . A -colouring of
is called a -rs colouring of if there is no path in with
. For , the problem -STAR COLOURABILITY
takes a graph as input and asks whether admits a -star colouring.
The problem -RS COLOURABILITY is defined similarly. Recently, Brause et al.
(Electron. J. Comb., 2022) investigated the complexity of 3-star colouring with
respect to the graph diameter. We study the complexity of -star colouring
and -rs colouring with respect to the maximum degree for all . For
, let us denote the least integer such that -STAR COLOURABILITY
(resp. -RS COLOURABILITY) is NP-complete for graphs of maximum degree by
(resp. ).
We prove that for and , -STAR COLOURABILITY is NP-complete
for graphs of maximum degree . We also show that -RS COLOURABILITY is
NP-complete for planar 3-regular graphs of girth 5 and -RS COLOURABILITY is
NP-complete for triangle-free graphs of maximum degree for .
Using these results, we prove the following: (i) for and ,
-STAR COLOURABILITY is NP-complete for -regular graphs if and only if
; and (ii) for , -RS COLOURABILITY is NP-complete
for -regular graphs if and only if
The Medieval Risk-Reward Society: Courts, Adventure, and Love in the European Middle Ages
Item embargoed for five year
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