eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-08-21
61:1
61:14
10.4230/LIPIcs.MFCS.2023.61
article
Parameterized Complexity of Domination Problems Using Restricted Modular Partitions
Lafond, Manuel
1
https://orcid.org/0000-0002-5305-7372
Luo, Weidong
1
https://orcid.org/0009-0003-5300-606X
Department of Computer Science, Université de Sherbrooke, Canada
For a graph class 𝒢, we define the 𝒢-modular cardinality of a graph G as the minimum size of a vertex partition of G into modules that each induces a graph in 𝒢. This generalizes other module-based graph parameters such as neighborhood diversity and iterated type partition. Moreover, if 𝒢 has bounded modular-width, the W[1]-hardness of a problem in 𝒢-modular cardinality implies hardness on modular-width, clique-width, and other related parameters. Several FPT algorithms based on modular partitions compute a solution table in each module, then combine each table into a global solution. This works well when each table has a succinct representation, but as we argue, when no such representation exists, the problem is typically W[1]-hard. We illustrate these ideas on the generic (α, β)-domination problem, which is a generalization of known domination problems such as Bounded Degree Deletion, k-Domination, and α-Domination. We show that for graph classes 𝒢 that require arbitrarily large solution tables, these problems are W[1]-hard in the 𝒢-modular cardinality, whereas they are fixed-parameter tractable when they admit succinct solution tables. This leads to several new positive and negative results for many domination problems parameterized by known and novel structural graph parameters such as clique-width, modular-width, and cluster-modular cardinality.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol272-mfcs2023/LIPIcs.MFCS.2023.61/LIPIcs.MFCS.2023.61.pdf
modular-width
parameterized algorithms
W-hardness
𝒢-modular cardinality