Counting list matrix partitions of graphs
Andreas G\"obel, Leslie Ann Goldberg, Colin McQuillan, David, Richerby, Tomoyuki Yamakami

TL;DR
This paper characterizes the computational complexity of counting list M-partitions in graphs, providing a polynomial-time algorithm for certain matrices and proving #P-completeness for others, with implications for graph structures and restrictions.
Contribution
It introduces a dichotomy theorem for counting list M-partitions, identifying structural conditions for tractability and hardness, and develops algorithms using data structures like sparse-dense partitions.
Findings
Polynomial-time counting algorithms for matrices with derectangularising sequences
#P-completeness results for matrices without derectangularising sequences
NP-completeness of determining derectangularising sequences
Abstract
Given a symmetric D*D matrix M over {0,1,*}, a list M-partition of a graph G is a partition of G's vertices into D parts associated with the rows of M. The part of each vertex is chosen from a given list so that no edge of G maps to a 0 in M and no non-edge of G maps to a 1 in M. Many important graph-theoretic structures can be represented as list M-partitions, such as graph colourings, split graphs and homogeneous sets and pairs, which arise in the proofs of the weak and strong perfect graph conjectures. There has been quite a bit of work on determining for which matrices M computations involving list M-partitions are tractable. We focus on counting list M-partitions, given a graph G and a list for each vertex of G. We identify a set of "tractable" matrices and give an algorithm that counts list M-partitions in polynomial time for every (fixed) matrix M in this set. The algorithm uses…
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph Labeling and Dimension Problems · Graph theory and applications
