Edge Deletion Problem Research Articles

An FPT Algorithm for Directed Co-Graph Edge Deletion

In the directed co-graph edge-deletion problem, we are given a directed graph and an integer k, and the question is whether we can delete, at most, k edges so that the resulting graph is a directed co-graph. In this paper, we make two minor contributions. Firstly, we show that the problem is NP-hard. Then, we show that directed co-graphs are fully characterized by eight forbidden structures, each having, at most, six edges. Based on the symmetry properties and several refined observations, we develop a branching algorithm with a running time of O(2.733k), which is significantly more efficient compared to the brute-force algorithm, which has a running time of O(6k).

On the parameterized complexity of s-club cluster deletion problems

We study the parameterized complexity of the s-Club Cluster Edge Deletion (s-Club Cluster Vertex Deletion) problem: Given a graph G and two integers s≥2 and k≥1, is it possible to remove at most k edges (vertices) from G such that each connected component of the resulting graph has diameter at most s? Both s-Club Cluster Edge Deletion and s-Club Cluster Vertex Deletion problems are known to be NP-hard already when s=2. We prove that they admit a fixed-parameter tractable algorithm when parameterized by s and the treewidth of the input graph. The proof is based on a unified algorithm that solves the more general problem in which both edges and vertices can be removed from the input graph to obtain a set of disjoint components with bounded diameter. Our approach can also be exploited to solve a related problem, namely s-Club Cover, which asks whether it is possible to cover the vertices of a graph with at most d different s-clubs, for some fixed d≥1 and s≥2.

An improved fixed-parameter algorithm for 2-Club Cluster Edge Deletion

A 2-club is a graph of diameter at most two. In the decision version of the 2-Club Cluster Edge Deletion problem, an undirected graph G is given along with an integer k≥0 as parameter, and the question is whether G can be transformed into a disjoint union of 2-clubs by deleting at most k edges. A simple fixed-parameter algorithm solves the problem in O⁎(3k), and a decade-old algorithm improved this running to O⁎(2.74k) time via a more sophisticated case analysis. Unfortunately, this latter algorithm is shown to have a flawed branching scenario. A fixed-parameter algorithm that breaks the 3k barrier is presented in this paper, with a running time in O⁎(2.692k). This also improves the previously claimed running time.

Further parameterized algorithms for the [formula omitted]-free edge deletion problem

Given a graph G=(V,E) and a set F of forbidden subgraphs, we study the F-Free Edge Deletion problem, where the goal is to remove a minimum number of edges such that the resulting graph does not contain any F∈F as a (not necessarily induced) subgraph. Enright and Meeks (Algorithmica, 2018) gave an algorithm to solve F-Free Edge Deletion whose running time on an n-vertex graph G of treewidth tw(G) is bounded by 2O(|F|tw(G)r)n, if every graph in F has at most r vertices. We complement this result by showing that F-Free Edge Deletion is W[1]-hard when parameterized by tw(G)+|F|. We also show that F-Free Edge Deletion is W[2]-hard when parameterized by the combined parameters solution size, the feedback vertex set number and pathwidth of the input graph. A special case of particular interest is the situation in which F is the set Th+1 of all trees on h+1 vertices, so that we delete edges in order to obtain a graph in which every component contains at most h vertices. This is desirable from the point of view of restricting the spread of a disease in transmission networks [5]. We prove that Th+1-Free Edge Deletion is fixed-parameter tractable (FPT) when parameterized by the vertex cover number of the input graph. We also prove that it admits a kernel with 2kh vertices and 2kh2+k edges, when parameterized by k+h.

Improved kernel and algorithm for claw and diamond free edge deletion based on refined observations

In the ▪-Free Edge Deletion problem (CDFED), we are given a graph G and an integer k>0, and the question is whether there are at most k edges whose deletion results in a graph without claws and diamonds as induced subgraphs. Based on some refined observations, we propose a kernel of O(k3) vertices and O(k4) edges, significantly improving the previous kernel of O(k12) vertices and O(k24) edges. In addition, we derive an O⁎(3.792k)-time algorithm for CDFED.

A Polynomial Kernel for Diamond-Free Editing

Given a fixed graph H, the H-free editing problem asks whether we can edit at most k edges to make a graph contain no induced copy of H. We obtain a polynomial kernel for this problem when H is a diamond. The incompressibility dichotomy for H being a 3-connected graph and the classical complexity dichotomy suggest that except for H being a complete/empty graph, H-free editing problems admit polynomial kernels only for a few small graphs H. Therefore, we believe that our result is an essential step toward a complete dichotomy on the compressibility of H-free editing. Additionally, we give a cubic-vertex kernel for the diamond-free edge deletion problem, which is far simpler than the previous kernel of the same size for the problem.

An improved branching algorithm for the proper interval edge deletion problem

An improved branching algorithm for the proper interval edge deletion problem

Polynomial kernels for paw-free edge modification problems

Given a graph G and an integer k, the paw-free completion problem asks whether it is possible to add at most k edges to G to make it paw-free. The paw-free edge deletion problem is defined analogously. Sandeep and Sivadasan (IPEC 2015) asked whether these problems admit polynomial kernels. We answer both questions affirmatively by presenting, respectively, O(k)-vertex and O(k4)-vertex kernels for them. This is part of an ongoing program that aims at understanding the compressibility of H-free edge modification problems for general H.

Online Node- and Edge-Deletion Problems with Advice

In online edge- and node-deletion problems the input arrives node by node and an algorithm has to delete nodes or edges in order to keep the input graph in a given graph class Pi at all times. We consider only hereditary properties Pi , for which optimal online algorithms exist and which can be characterized by a set of forbidden subgraphs {{mathcal{F}}} and analyze the advice complexity of getting an optimal solution. We give almost tight bounds on the Delayed Connected{{mathcal{F}}}-Node-Deletion Problem, where all graphs of the family {mathcal{F}} have to be connected and almost tight lower and upper bounds for the DelayedH-Node-Deletion Problem, where there is one forbidden induced subgraph H that may be connected or not. For the DelayedH-Node-Deletion Problem the advice complexity is basically an easy function of the size of the biggest component in H. Additionally, we give tight bounds on the Delayed Connected{mathcal{F}}-Edge-Deletion Problem, where we have an arbitrary number of forbidden connected graphs. For the latter result we present an algorithm that computes the advice complexity directly from {mathcal{F}}. We give a separate analysis for the Delayed ConnectedH-Edge-Deletion Problem, which is less general but admits a bound that is easier to compute.

Deleting edges to restrict the size of an epidemic in temporal networks

Spreading processes on graphs are a natural model for a wide variety of real-world phenomena, including information spread over social networks and biological diseases spreading over contact networks. Often, the networks over which these processes spread are dynamic in nature, and can be modelled with temporal graphs. Here, we study the problem of deleting edges from a given temporal graph in order to reduce the number of vertices (temporally) reachable from a given starting point. This could be used to control the spread of a disease, rumour, etc. in a temporal graph. In particular, our aim is to find a temporal subgraph in which a process starting at any single vertex can be transferred to only a limited number of other vertices using a temporally-feasible path. We introduce a natural edge-deletion problem for temporal graphs and provide positive and negative results on its computational complexity and approximability.

Linear Kernels for Edge Deletion Problems to Immersion-Closed Graph Classes

Suppose ${\mathcal{F}}$ is a finite family of graphs. We consider the following meta-problem, called $\mathcal{F}$-Immersion Deletion: given a graph $G$ and integer $k$, decide whether the deletion...

Kernel for Kt-free Edge Deletion

In the Kt-free Edge Deletion problem the input is a graph G and an integer k, and the goal is to decide whether there is a set of at most k edges of G whose removal results in a graph with no clique of size t. In this paper we give a kernel to this problem with O(kt−1) vertices and edges.

Tractability of König edge deletion problems

A graph is said to be a König graph if the size of its maximum matching is equal to the size of its minimum vertex cover. The König Edge Deletion problem asks if in a given graph there exists a set of at most k edges whose deletion results in a König graph. While the vertex version of the problem (König vertex deletion) has been shown to be fixed-parameter tractable more than a decade ago, the fixed-parameter-tractability of the König Edge Deletion problem has been open since then, and has been conjectured to be W[1]-hard in several papers. In this paper, we settle the conjecture by proving it W[1]-hard. We prove that a variant of this problem, where we are given a graph G and a maximum matching M and we want a k-sized König edge deletion set that is disjoint from M, is fixed-parameter-tractable.

Optimal Surveillance of Covert Networks by Minimizing Inverse Geodesic Length

The inverse geodesic length (IGL) is a well-known and widely used measure of network performance. It equals the sum of the inverse distances of all pairs of vertices. In network analysis, IGL of a network is often used to assess and evaluate how well heuristics perform in strengthening or weakening a network. We consider the edge-deletion problem MINIGLED. Formally, given a graph G, a budget k, and a target inverse geodesic length T, the question is whether there exists a subset of edges X with |X| ≤ ck, such that the inverse geodesic length of G − X is at most T.In this paper, we design algorithms and study the complexity of MINIGL-ED. We show that it is NP-complete and cannot be solved in subexponential time even when restricted to bipartite or split graphs assuming the Exponential Time Hypothesis. In terms of parameterized complexity, we consider the problem with respect to various parameters. We show that MINIGL-ED is fixed-parameter tractable for parameter T and vertex cover by modeling the problem as an integer quadratic program. We also provide FPT algorithms parameterized by twin cover and neighborhood diversity combined with the deletion budget k. On the negative side we show that MINIGL-ED is W[1]-hard for parameter tree-width.

Parameterized complexity of fair deletion problems

Edge deletion problems are those where the goal is to find a subset of edges such that after its removal the graph satisfies the given graph property. Typically, we want to minimize the number of elements removed. In fair deletion problems, the objective is changed, so the maximum number of deletions in a neighborhood of a single vertex is minimized.We study the parameterized complexity of fair deletion problems concerning the structural parameters such as the tree-width, the path-width, the tree-depth, the size of minimum feedback vertex set, the neighborhood diversity, and the size of minimum vertex cover of graph G.We prove the W[1]-hardness of the fair FO vertex-deletion problem with respect to the combined size of the tree-depth and the minimum feedback vertex set number. Moreover, we show that there is no algorithm for fair FO vertex-deletion problem running in time no(k3), where n is the size of the graph and k is the sum of the mentioned parameters, provided that the Exponential Time Hypothesis holds.On the other hand, we present an FPT algorithm for the fair MSO edge-deletion problem parameterized by the size of minimum vertex cover and an FPT algorithm for the fair MSO vertex-deletion problem parameterized by the neighborhood diversity.

Triangle edge deletion on planar glasses-free RGB-digraphs

In this paper, we revisit the triangle edge deletion problem a variant of feedback arc set problem which is one of the Karp's 21 NP-complete problems. Given a graph G(V,E), it is to find the edge set F such that |F|≤k and G′(V,E∖F) is a triangle-free graph. Yannakakis [1] showed it is hard for simple digraphs as well. The latest results showed that triangle edge deletion problem is still hard even for (a) planar undirected graphs of maximum degree seven [2] and (b) RGB-digraphs [3] which is a special class of digraphs. To make a further step, we show a new result in this paper that triangle edge deletion problem is still NP-complete even for planar RGB-digraphs of maximum degree eight with no glasses. On the other hand, we can find a kernel consisting of 11k/3 vertices for planar RGB-digraphs with no glasses. The results have an interesting application for the derivation of a tighter dichotomy on resilience decision problem in database theory.

Finding even subgraphs even faster

In the Undirected Eulerian Edge Deletion problem, we are given an undirected graph and an integer k, and the objective is to delete k edges such that the resultant graph is a connected graph in which all the vertices have even degrees. The corresponding problem in digraphs where the resulting graph should be strongly connected and every vertex should have the same in-degree as its out-degree is called Directed Eulerian Edge Deletion. In this paper, using the technique of computing representative families of co-graphic matroids we design algorithms which solve these problems in time 2O(k)nO(1), improving the algorithms by Cygan et al. [Algorithmica, 2014] and affirmatively answer the open problem posed by them. The crucial insight we bring to these problems is to view the solution as an independent set of a co-graphic matroid.

Parameterized algorithms for Edge Biclique and related problems

Maximum Edge Biclique and related problems have wide applications in management science, bioinformatics, etc. In this paper, we study parameterized algorithms for Parameterized Edge Biclique problem, Parameterized Edge Biclique Packing problem, Parameterized Biclique Edge Deletion problem, and Parameterized Bipartite Biclique Clustering problem. For the Parameterized Edge Biclique problem, the current best result is of running time O⁎(2k), and we give a parameterized algorithm of running time O(nk2.5k⌈k⌉), where k is the parameter and n is the number of vertices in the given graph. For the Parameterized Edge Biclique Packing problem, based on randomized divide-and-conquer technique, a parameterized algorithm of running time O⁎(k⌈k⌉logk4(2k−1)t)) is given, where k is the parameter and t is the number of bicliques in the solution. We study the Parameterized Biclique Edge Deletion problem on bipartite graphs and general graphs, and give parameterized algorithms of running time O⁎(2k) and O⁎(3k), respectively. For the Parameterized Bipartite Biclique Clustering problem, based on modular decomposition method, a kernel of size O(k2) and a parameterized algorithm of running time O(2.42k(n+m)) are presented, where k is the parameter, n is the number of vertices, and m is the number of edges in the given graph.

Dealing with several parameterized problems by random methods

In this paper, we apply random methods to deal with several parameterized problems. For the Parameterized Weighted P3-Packing problem, by randomly partitioning the vertices in given graph, a tripartite graph can be obtained. We prove that the Parameterized Weighted P3-Packing problem can be solved in polynomial time on tripartite graphs. Based on the algorithm on tripartite graphs, a randomized parameterized algorithm of running time O⁎(32k) is given for the Parameterized Weighted P3-Packing problem. For the Parameterized Weighted Load Coloring problem, by randomly partitioning the vertices in given graph into two parts and studying the structure properties of the connected components in two parts, a randomized parameterized algorithm of running time O⁎(11.32k) is presented. For the Parameterized Claw-free Edge Deletion problem on Diamond-free Graphs, by combining random with branching methods, a parameterized algorithm of running time O⁎(2.42k) is given.

Unit interval editing is fixed-parameter tractable

Given a graph G and integers k1, k2, and k3, the unit interval editing problem asks whether G can be transformed into a unit interval graph by at most k1 vertex deletions, k2 edge deletions, and k3 edge additions. We give an algorithm solving this problem in time 2O(klog⁡k)⋅(n+m), where k:=k1+k2+k3, and n,m denote respectively the numbers of vertices and edges of G. Therefore, it is fixed-parameter tractable parameterized by the total number of allowed operations.Our algorithm implies the fixed-parameter tractability of the unit interval edge deletion problem, for which we also present a more efficient algorithm running in time O(4k⋅(n+m)). Another result is an O(6k⋅(n+m))-time algorithm for the unit interval vertex deletion problem, significantly improving the algorithm of van 't Hof and Villanger, which runs in time O(6k⋅n6).