On Making a Distinguished Vertex of Minimum Degree by Vertex Deletion (original) (raw)

Graph and Election Problems Parameterized by Feedback Set Numbers

This work investigates the parameterized complexity of three related graph modification problems. Given a directed graph, a distinguished vertex, and a positive integer k, Minimum Indegree Deletion asks for a vertex subset of size at most k whose removal makes the distinguished vertex the only vertex with minimum indegree. Minimum Degree Deletion is analogously defined, but deals with undirected graphs. Bounded Degree Deletion is also defined on undirected graphs, but has a positive integer d instead of a distinguished vertex as part of the input. It asks for a vertex subset of size at most k whose removal results in a graph in which every vertex has degree at most d. The first two problems have applications in computational social choice whereas the third problem is used in computational biology. We investigate the parameterized complexity with respect to the parameters "treewidth", "size of a feedback vertex set" and "size of a feedback edge set" resp...

On the Parameterized Complexity of Simultaneous Deletion Problems

2017

For a family of graphs F, an n-vertex graph G, and a positive integer k, the F-Deletion problem asks whether we can delete at most k vertices from G to obtain a graph in F. F-Deletion generalizes many classical graph problems such as Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. A (multi) graph G = (V, \cup_{i=1}^{\alpha} E_{i}), where the edge set of G is partitioned into \alpha color classes, is called an \alpha-edge-colored graph. A natural extension of the F-Deletion problem to edge-colored graphs is the Simultaneous (F_1, \ldots, F_\alpha)-Deletion problem. In the latter problem, we are given an \alpha-edge-colored graph G and the goal is to find a set S of at most k vertices such that each graph G_i - S, where G_i = (V, E_i) and 1 \leq i \leq \alpha, is in F_i. Recently, a subset of the authors considered the aforementioned problem with F_1 = \ldots = F_\alpha being the family of all forests. They showed that the problem is fixed-parameter tractable when parame...

A Polynomial Kernel for Deletion to Ptolemaic Graphs

2021

For a family of graphs F , given a graph G and an integer k, the F-Deletion problem asks whether we can delete at most k vertices from G to obtain a graph in the family F . The F-Deletion problems for all non-trivial families F that satisfy the hereditary property on induced subgraphs are known to be NP-hard by a result of Yannakakis (STOC’78). Ptolemaic graphs are the graphs that satisfy the Ptolemy inequality, and they are the intersection of chordal graphs and distance-hereditary graphs. Equivalently, they form the set of graphs that do not contain any chordless cycles or a gem as an induced subgraph. (A gem is the graph on 5 vertices, where four vertices form an induced path, and the fifth vertex is adjacent to all the vertices of this induced path.) The Ptolemaic Deletion problem is the F-Deletion problem, where F is the family of Ptolemaic graphs. In this paper we study Ptolemaic Deletion from the viewpoint of Kernelization Complexity, and obtain a kernel with O(k6) vertices f...