Abstract
Graph Theory We study partitions of the vertex set of a given graph into cells that each induce a subgraph in a given family, and for which edges can have ends in different cells only when those cells correspond to adjacent vertices of a fixed template graph H. For triangle-free templates, a general collection of graph families for which the partitioning problem can be solved in polynomial time is described. For templates with a triangle, the problem is in some cases shown to be NP-complete.
Highlights
A k-colouring of a graph G is a partition of V (G) into k independent sets, some of which may be empty
If H is a graph with vertex set {v1, v2, . . . , vk}, a homomorphism of G to H is an ordered partition (V1, V2, . . . , Vk) of V (G), with some cells allowed to be empty, such that an edge of G can have one end in Vi and the other in Vj only if vivj ∈ E(H)
We prove that H having a triangle is the “dividing line” for NP-completeness of some large classes of conditional colouring problems with starter-matrix condition C and template H
Summary
A k-colouring of a graph G is a partition of V (G) into k independent sets, some of which may be empty. When C is the collection of non-null connected graphs, and it is required that there be an edge of G with ends in different cells whenever these cells correspond to adjacent vertices of H, conditional colouring with condition C and template H becomes contractability to H [6, 24]. The input graph is partitioned into cells whose structure is determined by the entries of a given matrix M , as is the structure of the edges with ends in different cells We borrow this idea to help describe our general collection of graph families . The conditions considered in this paper are not additive
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