Core Concepts
This paper presents efficient kernelization algorithms for modifying a given graph to have at most two distinct eigenvalues by vertex deletion, edge editing, edge deletion, and edge addition.
Abstract
The paper studies the problem of modifying a given graph G = (V, E) such that the resulting graph has at most two distinct eigenvalues. This is equivalent to transforming G into a collection of equal-sized cliques. The authors consider four different graph modification operations: vertex deletion (2-Eigenvalue Vertex Deletion, 2-EVD), edge editing (2-Eigenvalue Edge Editing, 2-EEE), edge deletion (2-Eigenvalue Edge Deletion, 2-EED), and edge addition (2-Eigenvalue Edge Addition, 2-EEA).
For the 2-EVD problem, the authors provide a kernel of size O(k^3), where k is the solution size (the number of vertices to be deleted). For the 2-EEE and 2-EED problems, they provide kernels of size O(k^2). Finally, for the 2-EEA problem, they provide a linear kernel of size 6k.
The key steps in the kernelization algorithms include:
Identifying a maximal set of vertex-disjoint induced P3s (paths on 3 vertices) in the graph, as a graph is a cluster graph if and only if it does not contain an induced P3.
Applying various reduction rules to bound the number of cliques in the graph based on their sizes and the vertices' neighborhoods.
Exploiting the fact that in a yes-instance, the vertices in the graph can be partitioned into two sets: those with degree equal to the average degree, and those with degree not equal to the average degree.
The authors also discuss the connections between their results and previous work on related problems, such as Balanced Cluster Completion and s-Club Cluster Vertex Deletion.
Stats
The paper does not contain any explicit numerical data or statistics. The focus is on developing efficient kernelization algorithms for the considered graph modification problems.
Quotes
There are no direct quotes from the paper that are particularly striking or support the key logics.