核心概念
This paper presents polynomial-delay enumeration kernelizations for enumerating all degree-d cuts, all minimal degree-d cuts, and all maximal degree-d cuts in a graph, parameterized by the vertex cover number, neighborhood diversity, and clique partition number of the input graph.
要約
The paper considers the Degree-d-Cut problem, which asks whether a given graph has a d-cut, where a d-cut is a bipartition of the vertices such that every vertex in one part has at most d neighbors in the other part. The authors study three enumeration variants of this problem:
- Enum Deg-d-Cut: Enumerate all the d-cuts of the graph.
- Enum Min Deg-d-Cut: Enumerate all the minimal d-cuts of the graph.
- Enum Max Deg-d-Cut: Enumerate all the maximal d-cuts of the graph.
The authors provide the following results:
- For Enum Min Deg-d-Cut parameterized by the vertex cover number (vc), they give a fully-polynomial enumeration kernel of size O(d^3 * vc^(d+1)).
- For Enum Deg-d-Cut and Enum Max Deg-d-Cut parameterized by vc, they give polynomial-delay enumeration kernels of size O(d^3 * vc^(d+1)).
- For all three variants parameterized by the neighborhood diversity (nd), they give polynomial-delay enumeration kernels of size O(d^2 * nd).
- For all three variants parameterized by the clique partition number (pc), they give bijective enumeration kernelizations of size O(pc^(d+2)).
The authors use structural properties of d-cuts and the marking scheme to establish these results. They also show that these problems do not admit polynomial-delay enumeration kernels of polynomial size when parameterized by treewidth or cliquewidth, unless NP ⊆ coNP/poly.