36 research outputs found
A Brightwell-Winkler type characterisation of NU graphs
In 2000, Brightwell and Winkler characterised dismantlable graphs as the
graphs for which the Hom-graph , defined on the set of
homomorphisms from to , is connected for all graphs . This shows that
the reconfiguration version of the -colouring
problem, in which one must decide for a given whether is
connected, is trivial if and only if is dismantlable.
We prove a similar starting point for the reconfiguration version of the
-extension problem. Where is the subgraph of the
Hom-graph induced by the -colourings extending the
-precolouring of , the reconfiguration version
of the -extension problem asks, for a given -precolouring of a graph
, if is connected. We show that the graphs for which
is connected for every choice of are exactly the
graphs. This gives a new characterisation of graphs, a
nice class of graphs that is important in the algebraic approach to the -dichotomy.
We further give bounds on the diameter of for
graphs , and show that shortest path between two vertices of can be found in parameterised polynomial time. We apply our
results to the problem of shortest path reconfiguration, significantly
extending recent results.Comment: 17 pages, 1 figur
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
Reconfiguring graph homomorphisms on the sphere
Given a loop-free graph H, the reconfiguration problem for homomorphisms to H (also called H-colourings) asks: given two H-colourings f of g of a graph G, is it possible to transform f into g by a sequence of single-vertex colour changes such that every intermediate mapping is an H-colouring? This problem is known to be polynomial-time solvable for a wide variety of graphs H (e.g. all C4-free graphs) but only a handful of hard cases are known. We prove that this problem is PSPACE-complete whenever H is a K2,3-free quadrangulation of the 2-sphere (equivalently, the plane) which is not a 4-cycle. From this result, we deduce an analogous statement for non-bipartite K2,3-free quadrangulations of the projective plane. This include several interesting classes of graphs, such as odd wheels, for which the complexity was known, and 4-chromatic generalized Mycielski graphs, for which it was not.
If we instead consider graphs G and H 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 H-colourings is PSPACE-complete whenever H is a reflexive K4-free triangulation of the 2-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 H-Recolouring is known to be PSPACE-complete for reflexive instances