Parameterized Complexity Classification of Deletion to List Matrix-Partition for Low-Order Matrices

Given a symmetric l x l matrix M=(m_{i,j}) with entries in {0,1,*}, a graph G and a function L : V(G) - > 2^{[l]} (where [l] = {1,2,...,l}), a list M-partition of G with respect to L is a partition of V(G) into l parts, say, V_1, V_2, ..., V_l such that for each i,j in {1,2,...,l}, (i) if m_{i,j}=0 then for any u in V_i and v in V_j, uv not in E(G), (ii) if m_{i,j}=1 then for any (distinct) u in V_i and v in V_j, uv in E(G), (iii) for each v in V(G), if v in V_i then i in L(v). We consider the Deletion to List M-Partition problem that takes as input a graph G, a list function L:V(G) - > 2^[l] and a positive integer k. The aim is to determine whether there is a k-sized set S subseteq V(G) such that G-S has a list M-partition. Many important problems like Vertex Cover, Odd Cycle Transversal, Split Vertex Deletion, Multiway Cut and Deletion to List Homomorphism are special cases of the Deletion to List M-Partition problem. In this paper, we provide a classification of the parameterized complexity of Deletion to List M-Partition, parameterized by k, (a) when M is of order at most 3, and (b) when M is of order 4 with all diagonal entries belonging to {0,1}

Counting list matrix partitions of graphs

a

Counting list matrix partitions of graphs

Given a symmetric D × D matrix M over {0, 1, ∗}, a list M-partition of a graph G is a partition of the vertices of G into D parts which are associated with the rows of M. The part of each vertex is chosen from a given list in such a way that no edge of G is mapped to a 0 in M and no non-edge of G is mapped to a 1 in M. Many important graph-theoretic structures can be represented as list M-partitions including graph colourings, split graphs and homogeneous sets and pairs, which arise in the proofs of the weak and strong perfect graph conjectures. Thus, there has been quite a bit of work on determining for which matrices M computations involving list M-partitions are tractable. This paper focuses on the problem of counting list M-partitions, given a graph G and given a list for each vertex of G. We identify a certain set of “tractable” matrices M. We give an algorithm that counts list M partitions in polynomial time for every (fixed) matrix M in this set. The algorithm relies on data structures such as sparse-dense partitions and subcube decompositions to reduce each problem instance to a sequence of problem instances in which the lists have a certain useful structure that restricts access to portions of M in which the interactions of 0s and 1s is controlled. We show how to solve the resulting restricted instances by converting them into particular counting constraint satisfaction problems (#CSPs) which we show how to solve using a constraint satisfaction technique known as “arc-consistency”. For every matrix M for which our algorithm fails, we show that the problem of counting list M-partitions is #P-complete. Furthermore, we give an explicit characterisation of the dichotomy theorem — counting list M partitions is tractable (in FP) if the matrix M has a structure called a derectangularising sequence. If M has no derectangularising sequence, we show that counting list M-partitions is #P-hard. We show that the meta-problem of determining whether a given matrix has a derectangularising sequence is NP-complete. Finally, we show that list M partitions can be used to encode cardinality restrictions in M partitions problems and we use this to give a polynomial-time algorithm for counting homogeneous pairs in graphs