TY - JOUR

T1 - Reconfiguring graph homomorphisms on the sphere

AU - Lee, Jae Baek

AU - Noel, Jonathan A.

AU - Siggers, Mark

N1 - Publisher Copyright:
© 2020 Elsevier Ltd

PY - 2020/5

Y1 - 2020/5

N2 - 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 the reconfiguration problem is known to be PSPACE-complete for reflexive instances.

AB - 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 the reconfiguration problem is known to be PSPACE-complete for reflexive instances.

UR - http://www.scopus.com/inward/record.url?scp=85078838529&partnerID=8YFLogxK

U2 - 10.1016/j.ejc.2020.103086

DO - 10.1016/j.ejc.2020.103086

M3 - Article

AN - SCOPUS:85078838529

SN - 0195-6698

VL - 86

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

M1 - 103086

ER -