27,437 research outputs found
Complexity of colouring problems restricted to unichord-free and \{square,unichord\}-free graphs
A \emph{unichord} in a graph is an edge that is the unique chord of a cycle.
A \emph{square} is an induced cycle on four vertices. A graph is
\emph{unichord-free} if none of its edges is a unichord. We give a slight
restatement of a known structure theorem for unichord-free graphs and use it to
show that, with the only exception of the complete graph , every
square-free, unichord-free graph of maximum degree~3 can be total-coloured with
four colours. Our proof can be turned into a polynomial time algorithm that
actually outputs the colouring. This settles the class of square-free,
unichord-free graphs as a class for which edge-colouring is NP-complete but
total-colouring is polynomial
The complexity of Free-Flood-It on 2xn boards
We consider the complexity of problems related to the combinatorial game
Free-Flood-It, in which players aim to make a coloured graph monochromatic with
the minimum possible number of flooding operations. Our main result is that
computing the length of an optimal sequence is fixed parameter tractable (with
the number of colours present as a parameter) when restricted to rectangular
2xn boards. We also show that, when the number of colours is unbounded, the
problem remains NP-hard on such boards. This resolves a question of Clifford,
Jalsenius, Montanaro and Sach (2010)
The Complexity of Change
Many combinatorial problems can be formulated as "Can I transform
configuration 1 into configuration 2, if certain transformations only are
allowed?". An example of such a question is: given two k-colourings of a graph,
can I transform the first k-colouring into the second one, by recolouring one
vertex at a time, and always maintaining a proper k-colouring? Another example
is: given two solutions of a SAT-instance, can I transform the first solution
into the second one, by changing the truth value one variable at a time, and
always maintaining a solution of the SAT-instance? Other examples can be found
in many classical puzzles, such as the 15-Puzzle and Rubik's Cube.
In this survey we shall give an overview of some older and more recent work
on this type of problem. The emphasis will be on the computational complexity
of the problems: how hard is it to decide if a certain transformation is
possible or not?Comment: 28 pages, 6 figure
A Study of -dipath Colourings of Oriented Graphs
We examine -colourings of oriented graphs in which, for a fixed integer , vertices joined by a directed path of length at most must be
assigned different colours. A homomorphism model that extends the ideas of
Sherk for the case is described. Dichotomy theorems for the complexity of
the problem of deciding, for fixed and , whether there exists such a
-colouring are proved.Comment: 14 page
Reconfiguring Graph Homomorphisms on the Sphere
Given a loop-free graph , the reconfiguration problem for homomorphisms to
(also called -colourings) asks: given two -colourings of of a
graph , is it possible to transform into by a sequence of
single-vertex colour changes such that every intermediate mapping is an
-colouring? This problem is known to be polynomial-time solvable for a wide
variety of graphs (e.g. all -free graphs) but only a handful of hard
cases are known. We prove that this problem is PSPACE-complete whenever is
a -free quadrangulation of the -sphere (equivalently, the plane)
which is not a -cycle. From this result, we deduce an analogous statement
for non-bipartite -free quadrangulations of the projective plane. This
include several interesting classes of graphs, such as odd wheels, for which
the complexity was known, and -chromatic generalized Mycielski graphs, for
which it was not.
If we instead consider graphs and with loops on every vertex (i.e.
reflexive graphs), then the reconfiguration problem is defined in a similar way
except that a vertex can only change its colour to a neighbour of its current
colour. In this setting, we use similar ideas to show that the reconfiguration
problem for -colourings is PSPACE-complete whenever is a reflexive
-free triangulation of the -sphere which is not a reflexive triangle.
This proof applies more generally to reflexive graphs which, roughly speaking,
resemble a triangulation locally around a particular vertex. This provides the
first graphs for which -Recolouring is known to be PSPACE-complete for
reflexive instances.Comment: 22 pages, 9 figure
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