Polynomial kernels for 3-leaf power graph modification problems

A graph G=(V,E) is a 3-leaf power iff there exists a tree T whose leaves are V and such that (u,v) is an edge iff u and v are at distance at most 3 in T. The 3-leaf power graph edge modification problems, i.e. edition (also known as the closest 3-leaf power), completion and edge-deletion, are FTP when parameterized by the size of the edge set modification. However polynomial kernel was known for none of these three problems. For each of them, we provide cubic kernels that can be computed in linear time for each of these problems. We thereby answer an open problem first mentioned by Dom, Guo, Huffner and Niedermeier (2005).Comment: Submitte

Lower Bounds for Leaf Rank of Leaf Powers

Leaf powers and $k$-leaf powers have been studied for over 20 years, but there are still several aspects of this graph class that are poorly understood. One such aspect is the leaf rank of leaf powers, i.e. the smallest number $k$ such that a graph $G$ is a $k$-leaf power. Computing the leaf rank of leaf powers has proved a hard task, and furthermore, results about the asymptotic growth of the leaf rank as a function of the number of vertices in the graph have been few and far between. We present an infinite family of rooted directed path graphs that are leaf powers, and prove that they have leaf rank exponential in the number of vertices (utilizing a type of subtree model first presented by Rautenbach [Some remarks about leaf roots. Discrete mathematics, 2006]). This answers an open question by Brandst\"adt et al. [Rooted directed path graphs are leaf powers. Discrete mathematics, 2010].Comment: Submitted to IWOCA 2024. 14 pages, 5 figure

The Complexity of Cluster Vertex Splitting and Company

Clustering a graph when the clusters can overlap can be seen from three different angles: We may look for cliques that cover the edges of the graph, we may look to add or delete few edges to uncover the cluster structure, or we may split vertices to separate the clusters from each other. Splitting a vertex $v$ means to remove it and to add two new copies of $v$ and to make each previous neighbor of $v$ adjacent with at least one of the copies. In this work, we study the underlying computational problems regarding the three angles to overlapping clusterings, in particular when the overlap is small. We show that the above-mentioned covering problem, which also has been independently studied in different contexts,is NP-complete. Based on a previous so-called critical-clique lemma, we leverage our hardness result to show that Cluster Editing with Vertex Splitting is also NP-complete, resolving an open question by Abu-Khzam et al. [ISCO 2018]. We notice, however, that the proof of the critical-clique lemma is flawed and we give a counterexample. Our hardness result also holds under a version of the critical-clique lemma to which we currently do not have a counterexample. On the positive side, we show that Cluster Vertex Splitting admits a vertex-linear problem kernel with respect to the number of splits.Comment: 30 pages, 9 figure

Pairwise Compatibility Graphs: A Survey

International audienceA graph $G=(V,E)$ is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree $T$ and two nonnegative real numbers $d_{min}$ and $d_{max}$ such that each leaf $u$ of $T$ is a node of $V$ and there is an edge $(u,v) \in E$ if and only if $d_{min} \leq d_T (u, v) \leq d_{max}$, where $d_T (u, v)$ is the sum of weights of the edges on the unique path from $u$ to $v$ in $T$. In this article, we survey the state of the art concerning this class of graphs and some of its subclasses

Kernelization for Balanced Graph Clustering

The problems of Balanced Graph Clustering ask whether it is possible to modify a graph such that it becomes a cluster graph where no cluster has a size larger than a given multiplicative factor or absolute difference relative to any other cluster in the graph, by at most k graph modifications. In this thesis we study the problems with respect to the graph modification operations vertex deletion, edge addition and edge deletion. We will show NP-completeness and give polynomial kernels for each version.Masteroppgave i informatikkINF399MAMN-INFMAMN-PRO

Graph Powers: Hardness Results, Good Characterizations and Efficient Algorithms

Given a graph H = (V_H,E_H) and a positive integer k, the k-th power of H, written H^k, is the graph obtained from H by adding edges between any pair of vertices at distance at most k in H; formally, H^k = (V_H, {xy | 1 <= d_H (x, y) <= k}). A graph G is the k-th power of a graph H if G = H^k, and in this case, H is a k-th root of G. Our investigations deal with the computational complexity of recognizing k-th powers of general graphs as well as restricted graphs. This work provides new NP-completeness results, good characterizations and efficient algorithms for graph powers