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In the $K_t$-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 a graph with no clique of size $t$. In this paper we give a kernel to this problem with $O(k^{t-1})$ vertices and edges.
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It is known that the problem of deleting at most k vertices to obtain a proper interval graph (Proper Interval Vertex Deletion) is fixed parameter tractable. However, whether the problem admits a polynomial kernel or not was open. Here, we answers this question in affirmative by obtaining a polynomial kernel for Proper Interval Vertex Deletion. This resolves an open question of van Bevern, Komusiewicz, Moser, and Niedermeier.
In the {claw, diamond}-free edge deletion problem, we are given a graph $G$ and an integer $k>0$, the question is whether there are at most $k$ edges whose deletion results in a graph without claws and diamonds as induced graphs. Based on some refined observations, we propose a kernel of $O(k^3)$ vertices and $O(k^4)$ edges, significantly improving the previous kernel of $O(k^{12})$ vertices and $O(k^{24})$ edges. In addition, we derive an $O^*(3.792^k)$-time algorithm for the {claw, diamond}-free edge deletion problem.
The Chordal Vertex Deletion (ChVD) problem asks to delete a minimum number of vertices from an input graph to obtain a chordal graph. In this paper we develop a polynomial kernel for ChVD under the parameterization by the solution size, as well as poly(opt) approximation algorithm. The first result answers an open problem of Marx from 2006 [WG 2006, LNCS 4271, 37-48].
For a set of graphs $\mathcal{H}$, the \textsc{$\mathcal{H}$-free Edge Deletion} problem asks to find whether there exist at most $k$ edges in the input graph whose deletion results in a graph without any induced copy of $H\in\mathcal{H}$. In \cite{cai1996fixed}, it is shown that the problem is fixed-parameter tractable if $\mathcal{H}$ is of finite cardinality. However, it is proved in \cite{cai2013incompressibility} that if $\mathcal{H}$ is a singleton set containing $H$, for a large class of $H$, there exists no polynomial kernel unless $coNP\subseteq NP/poly$. In this paper, we present a p…
The line graph of a graph $G$ is the graph $L(G)$ whose vertex set is the edge set of $G$ and there is an edge between $e,f\in E(G)$ if $e$ and $f$ share an endpoint in $G$. A graph is called line graph if it is a line graph of some graph. We study the Line-Graph-Edge Deletion problem, which asks whether we can delete at most $k$ edges from the input graph $G$ such that the resulting graph is a line graph. More precisely, we give a polynomial kernel for Line-Graph-Edge Deletion with $\mathcal{O}(k^{5})$ vertices. This answers an open question posed by Falk Hüffner at Workshop on Kernels (WorKe…
Kernelization of 2-Club Cluster Edge Deletion on Interval Graphs
本文针对区间图上2-团簇边删除问题，得到大小为O(k⁵)的多项式顶点核，还证明该问题在单位区间图上多项式可解、在分裂图上NP-难，回应了其多项式核的开放问题。
In Directed Feeback Arc Set (DFAS) we search for a set of at most $k$ arcs which intersect every cycle in the input digraph. It is a well-known open problem in parameterized complexity to decide if DFAS admits a kernel of polynomial size. We consider $\mathcal{C}$-Arc Deletion Set ($\mathcal{C}$-ADS), a variant of DFAS where we want to remove at most $k$ arcs from the input digraph in order to turn it into a digraph of a class $\mathcal{C}$. In this work, we choose $\mathcal{C}$ to be the class of funnels. Funnel-Arc Deletion Set is NP-hard even if the input is a DAG, but is fixed-parameter tr…
In the Proper Interval Vertex Deletion problem (PIVD for short), we are given a graph $G$ and an integer parameter $k>0$, and the question is whether there are at most $k$ vertices in $G$ whose removal results in a proper interval graph. It is known that the PIVD problem is fixed-parameter tractable and admits a polynomial but "unreasonably" large kernel of $O(k^{53})$ vertices. A natural question is whether the problem admits a polynomial kernel of "reasonable" size. In this paper, we answer this question by deriving an $O(k^7)$-vertex kernel for the PIVD problem. Our kernelization is based o…
Given a graph $G$ and a parameter $k$, the Chordal Vertex Deletion (CVD) problem asks whether there exists a subset $U\subseteq V(G)$ of size at most $k$ that hits all induced cycles of size at least 4. The existence of a polynomial kernel for CVD was a well-known open problem in the field of Parameterized Complexity. Recently, Jansen and Pilipczuk resolved this question affirmatively by designing a polynomial kernel for CVD of size $O(k^{161}\log^{58}k)$, and asked whether one can design a kernel of size $O(k^{10})$. While we do not completely resolve this question, we design a significantly …
A diamond is a graph obtained by removing an edge from a complete graph on four vertices. A graph is diamond-free if it does not contain an induced diamond. The Diamond-free Edge Deletion problem asks whether there exist at most $k$ edges in the input graph $G$ whose deletion results in a diamond-free graph. For this problem, a polynomial kernel of $O(k^4$) vertices was found by Fellows et. al. (Discrete Optimization, 2011). In this paper, we give an improved kernel of $O(k^3)$ vertices for Diamond-free Edge Deletion. Further, we give an $O(k^2)$ vertex kernel for a related problem {Diamond,K_…
In the $\mathcal{F}$-Minor-Free Deletion problem one is given an undirected graph $G$, an integer $k$, and the task is to determine whether there exists a vertex set $S$ of size at most $k$, so that $G-S$ contains no graph from the finite family $\mathcal{F}$ as a minor. It is known that whenever $\mathcal{F}$ contains at least one planar graph, then $\mathcal{F}$-Minor-Free Deletion admits a polynomial kernel, that is, there is a polynomial-time algorithm that outputs an equivalent instance of size $k^{\mathcal{O}(1)}$ [Fomin, Lokshtanov, Misra, Saurabh; FOCS 2012]. However, this result relie…
In an edge modification problem, we are asked to modify at most $k$ edges to a given graph to make the graph satisfy a certain property. Depending on the operations allowed, we have the completion problems and the edge deletion problems. A great amount of efforts have been devoted to understanding the kernelization complexity of these problems. We revisit several well-studied edge modification problems, and develop improved kernels for them: \begin{itemize} \item a $2 k$-vertex kernel for the cluster edge deletion problem, \item a $3 k^2$-vertex kernel for the trivially perfect completion prob…
A graph $G$ is called a \emph{block graph} if each maximal $2$-connected component of $G$ is a clique. In this paper we study the Block Graph Vertex Deletion from the perspective of fixed parameter tractable (FPT) and kernelization algorithm. In particular, an input to Block Graph Vertex Deletion consists of a graph $G$ and a positive integer $k$ and the objective to check whether there exists a subset $S \subseteq V(G)$ of size at most $k$ such that the graph induced on $V(G)\setminus S$ is a block graph. In this paper we give an FPT algorithm with running time $4^kn^{O(1)}$ and a polynomial …
A graph is distance-hereditary if for any pair of vertices, their distance in every connected induced subgraph containing both vertices is the same as their distance in the original graph. The Distance-Hereditary Vertex Deletion problem asks, given a graph $G$ on $n$ vertices and an integer $k$, whether there is a set $S$ of at most $k$ vertices in $G$ such that $G-S$ is distance-hereditary. This problem is important due to its connection to the graph parameter rank-width that distance-hereditary graphs are exactly graphs of rank-width at most $1$. Eiben, Ganian, and Kwon (MFCS' 16) proved tha…
For a graph $H$, the $H$-free Edge Deletion problem asks whether there exist at most $k$ edges whose deletion from the input graph $G$ results in a graph without any induced copy of $H$. We prove that $H$-free Edge Deletion is NP-complete if $H$ is a graph with at least two edges and $H$ has a component with maximum number of vertices which is a tree or a regular graph. Furthermore, we obtain that these NP-complete problems cannot be solved in parameterized subexponential time, i.e., in time $2^{o(k)}\cdot |G|^{O(1)}$, unless Exponential Time Hypothesis fails.
Polynomial Kernel and Incompressibility for Prison-Free Edge Deletion and Completion
本文研究无监狱边删除问题，证明其具有多项式核，反驳了Marx和Sandeep关于H-free边删除不可压缩性的猜想，同时指出监狱补图对应问题仍不可压缩。
In the Block Graph Deletion problem, we are given a graph $G$ on $n$ vertices and a positive integer $k$, and the objective is to check whether it is possible to delete at most $k$ vertices from $G$ to make it a block graph, i.e., a graph in which each block is a clique. In this paper, we obtain a kernel with $\mathcal{O}(k^{6})$ vertices for the Block Graph Deletion problem. This is a first step to investigate polynomial kernels for deletion problems into non-trivial classes of graphs of bounded rank-width, but unbounded tree-width. Our result also implies that Chordal Vertex Deletion admits …
Quadratic Kernel for Cliques or Trees Vertex Deletion
本文研究团或树顶点删除问题，将已有核大小从 $O(k^4)$ 改进到 $O(k^2)$，并首次将团或树顶点删除数作为结构参数，证明最长环问题在该参数下固定参数可解。
The pathwidth of a graph is a measure of how path-like the graph is. Given a graph G and an integer k, the problem of finding whether there exist at most k vertices in G whose deletion results in a graph of pathwidth at most one is NP- complete. We initiate the study of the parameterized complexity of this problem, parameterized by k. We show that the problem has a quartic vertex-kernel: We show that, given an input instance (G = (V, E), k); |V| = n, we can construct, in polynomial time, an instance (G', k') such that (i) (G, k) is a YES instance if and only if (G', k') is a YES instance, (ii)…
A Polynomial Kernel for Deletion to the Scattered Class of Cliques and Trees
本文提出了一种新的图删除问题，即删除后每个连通分量必须是 clique 或 tree，并给出了该问题的首个非平凡多项式核。
For a graph class $\mathcal{C}$, the $\mathcal{C}$-Edge-Deletion problem asks for a given graph $G$ to delete the minimum number of edges from $G$ in order to obtain a graph in $\mathcal{C}$. We study the $\mathcal{C}$-Edge-Deletion problem for $\mathcal{C}$ the permutation graphs, interval graphs, and other related graph classes. It follows from Courcelle's Theorem that these problems are fixed parameter tractable when parameterized by treewidth. In this paper, we present concrete FPT algorithms for these problems. By giving explicit algorithms and analyzing these in detail, we obtain algorit…
The unit interval vertex deletion problem asks for a set of at most $k$ vertices whose deletion from an $n$-vertex graph makes it a unit interval graph. We develop an $O(k^4)$-vertex kernel for the problem, significantly improving the $O(k^{53})$-vertex kernel of Fomin, Saurabh, and Villanger [ESA'12; SIAM J. Discrete Math 27(2013)]. We introduce a novel way of organizing cliques of a unit interval graph. Our constructive proof for the correctness of our algorithm, using interval models, greatly simplifies the destructive proofs, based on forbidden induced subgraphs, for similar problems in li…
Linear rankwidth is a linearized variant of rankwidth, introduced by Oum and Seymour [Approximating clique-width and branch-width. J. Combin. Theory Ser. B, 96(4):514--528, 2006]. Motivated from recent development on graph modification problems regarding classes of graphs of bounded treewidth or pathwidth, we study the Linear Rankwidth-1 Vertex Deletion problem (shortly, LRW1-Vertex Deletion). In the LRW1-Vertex Deletion problem, given an $n$-vertex graph $G$ and a positive integer $k$, we want to decide whether there is a set of at most $k$ vertices whose removal turns $G$ into a graph of lin…
In the Tree Deletion Set problem the input is a graph G together with an integer k. The objective is to determine whether there exists a set S of at most k vertices such that G-S is a tree. The problem is NP-complete and even NP-hard to approximate within any factor of OPT^c for any constant c. In this paper we give a O(k^4) size kernel for the Tree Deletion Set problem. To the best of our knowledge our result is the first counterexample to the "conventional wisdom" that kernelization algorithms automatically provide approximation algorithms with approximation ratio close to the size of the ke…
We study the CONNECTED η-TREEDEPTH DELETION problem where the input instance is an undireted graph G = (V, E) and an integer k. The objective is to decide if G has a set S \subseteq V(G) of at most k vertices such that G - S has treedepth at most ηand G[S] is connected. As this problem naturally generalizes the well-known CONNECTED VERTEX COVER, when parameterized by solution size k, the CONNECTED η-TREEDEPTH DELETION does not admit polynomial kernel unless NP \subseteq coNP/poly. This motivates us to design an approximate kernel of polynomial size for this problem. In this paper, we show that…
Let F be a fixed finite obstruction set of graphs and G be a graph revealed in an online fashion, node by node. The online Delayed F-Node-Deletion Problem (F-Edge-Deletion Problem}) is to keep G free of every H in F by deleting nodes (edges) until no induced subgraph isomorphic to any graph in F can be found in G. The task is to keep the number of deletions minimal. Advice complexity is a model in which an online algorithm has access to a binary tape of infinite length, on which an oracle can encode information to increase the performance of the algorithm. We are interested in the minimum numb…
In this paper we give a construction of cut sparsifiers of Benczur and Karger in the {\em dynamic} streaming setting in a single pass over the data stream. Previous constructions either required multiple passes or were unable to handle edge deletions. We use $\tilde{O}(1/\e^2)$ time for each stream update and $\tilde{O}(n/\e^2)$ time to construct a sparsifier. Our $\e$-sparsifiers have $O(n\log^3 n/\e^2)$ edges. The main tools behind our result are an application of sketching techniques of Ahn et al.[SODA'12] to estimate edge connectivity together with a novel application of sampling with limi…
The Treewidth-2 Vertex Deletion problem asks whether a set of at most $t$ vertices can be removed from a graph, such that the resulting graph has treewidth at most two. A graph has treewidth at most two if and only if it does not contain a $K_4$ minor. Hence, this problem corresponds to the NP-hard $\mathcal{F}$-Minor Cover problem with $\mathcal{F} = \{K_4\}$. For any variant of the $\mathcal{F}$-Minor Cover problem where $\mathcal{F}$ contains a planar graph, it is known that a polynomial kernel exists. I.e., a preprocessing routine that in polynomial time outputs an equivalent instance of s…
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 of at most $k$ edges of $G$ can result in a graph that does not contain any graph from $\mathcal{F}$ as an immersion. This problem is a close relative of the $\mathcal{F}$-Minor Deletion problem studied by Fomin et al. [FOCS 2012], where one deletes vertices in order to remove all minor models of graphs from $\mathcal{F}$. We prove that whenever all graphs from $\mathcal{F}$ are connected and at least…
For a fixed finite family of graphs $\mathcal{F}$, the $\mathcal{F}$-Minor-Free Deletion problem takes as input a graph $G$ and an integer $\ell$ and asks whether there exists a set $X \subseteq V(G)$ of size at most $\ell$ such that $G-X$ is $\mathcal{F}$-minor-free. For $\mathcal{F}=\{K_2\}$ and $\mathcal{F}=\{K_3\}$ this encodes Vertex Cover and Feedback Vertex Set respectively. When parameterized by the feedback vertex number of $G$ these two problems are known to admit a polynomial kernelization. Such a polynomial kernelization also exists for any $\mathcal{F}$ containing a planar graph b…
We introduce the s-Plex Cluster Vertex Deletion problem. Like the Cluster Vertex Deletion problem, it is NP-hard and motivated by graph-based data clustering. While the task in Cluster Vertex Deletion is to delete vertices from a graph so that its connected components become cliques, the task in s-Plex Cluster Vertex Deletion is to delete vertices from a graph so that its connected components become s-plexes. An s-plex is a graph in which every vertex is nonadjacent to at most s-1 other vertices; a clique is an 1-plex. In contrast to Cluster Vertex Deletion, s-Plex Cluster Vertex Deletion allo…
Let F be a finite set of graphs. In the F-Deletion problem, we are given an n-vertex graph G and an integer k as input, and asked whether at most k vertices can be deleted from G such that the resulting graph does not contain a graph from F as a minor. F-Deletion is a generic problem and by selecting different sets of forbidden minors F, one can obtain various fundamental problems such as Vertex Cover, Feedback Vertex Set or Treewidth t-Deletion. In this paper we obtain a number of generic algorithmic results about Planar F-Deletion, when F contains at least one planar graph. The highlights of…
A polynomial kernel for vertex deletion into bipartite permutation graphs
本文研究了二分排列顶点删除问题的核化，证明该NP完全问题存在具有$O(k^{62})$个顶点的多项式核。
Where Treewidth and Pathwidth Diverge: Towards a Uniform Kernel for Pathwidth-$η$ Deletion
该研究针对路径宽-η删除问题，打破树宽与路径宽表现相似的预期，证明其在三种参数化情形下存在统一多项式核，并推测以解大小k为参数时也存在统一核。
Given a graph $G=(V,E)$, a set $\mathcal{F}$ of forbidden subgraphs, we study $\mathcal{F}$-Free Edge Deletion, where the goal is to remove minimum number of edges such that the resulting graph does not contain any $F\in \mathcal{F}$ as a subgraph. For the parameter treewidth, the question of whether the problem is FPT has remained open. Here we give a negative answer by showing that the problem is W[1]-hard when parameterized by the treewidth, which rules out FPT algorithms under common assumption. Thus we give a solution to the conjecture posted by Jessica Enright and Kitty Meeks in [Algorit…
A Polynomial Kernel for Vertex Deletion to the Scattered Class of Proper Interval Graph and Trees
该研究针对（真区间图，树）-顶点删除问题，在其已被证明固定参数可处理的基础上，首次提出顶点规模为O(k^33)的非平凡多项式核。
The F-Minor-Free Deletion problem asks, for a fixed set F and an input consisting of a graph G and integer k, whether k vertices can be removed from G such that the resulting graph does not contain any member of F as a minor. This paper analyzes to what extent provably effective and efficient preprocessing is possible for F-Minor-Free Deletion. Fomin et al. (FOCS 2012) showed that the special case Planar F-Deletion (when F contains at least one planar graph) has a kernel of size f(F) * k^{g(F)} for some functions f and g. The degree g of the polynomial grows very quickly; it is not even known …
We study the problem of deleting the smallest set $S$ of vertices (resp. edges) from a given graph $G$ such that the induced subgraph (resp. subgraph) $G \setminus S$ belongs to some class $\mathcal{H}$. We consider the case where graphs in $\mathcal{H}$ have treewidth bounded by $t$, and give a general framework to obtain approximation algorithms for both vertex and edge-deletion settings from approximation algorithms for certain natural graph partitioning problems called $k$-Subset Vertex Separator and $k$-Subset Edge Separator, respectively. For the vertex deletion setting, our framework co…
The CONNECTED VERTEX COVER problem asks for a vertex cover in a graph that induces a connected subgraph. The problem is known to be fixed-parameter tractable (FPT), and is unlikely to have a polynomial sized kernel (under complexity theoretic assumptions) when parameterized by the solution size. In a recent paper, Lokshtanov et al.[STOC 2017], have shown an $α$-approximate kernel for the problem for every $α> 1$, in the framework of approximate or lossy kernelization. In this work, we exhibit lossy kernels and FPT algorithms for CONNECTED VERTEX COVER for parameters that are more natural and f…
Graph modification problems are typically asked as follows: is there a small set of operations that transforms a given graph to have a certain property. The most commonly considered operations include vertex deletion, edge deletion, and edge addition; for the same property, one can define significantly different versions by allowing different operations. We study a very general graph modification problem which allows all three types of operations: given a graph $G$ and integers $k_1$, $k_2$, and $k_3$, the \textsc{chordal editing} problem asks whether $G$ can be transformed into a chordal grap…
The problem of worst case edge deletion from a network is considered. Suppose that you have a communication network and you can delete a single edge. Which edge deletion causes the largest disruption? More formally, given a graph, which edge after deletion disconnects the maximum number of pairs of vertices, where ties for number of pairs disconnected are broken by finding an edge that increases the average shortest path length the maximum amount. This problem is interesting both practically and theoretically. We call it the \emph{single edge deletion problem}. Our contributions include formal…
A graph is said to be a Konig graph if the size of its maximum matching is equal to the size of its minimum vertex cover. The Konig Edge Deletion problem asks if in a given graph there exists a set of at most k edges whose deletion results in a Konig graph. While the vertex version of the problem (Konig vertex deletion) has been shown to be fixed-parameter tractable more than a decade ago, the fixed-parameter-tractability of the Konig 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…
Protrusion Decompositions Revisited: Uniform Lossy Kernels for Reducing Treewidth and Linear Kernels for Hitting Disconnected Minors
本文提出了一种新的核化算法，通过小损失实现Treewidth-d-删除问题的均匀核，同时在稀疏图类上获得线性核。
For a graph $H$, the $H$-free Edge Deletion problem asks whether there exist at most $k$ edges whose deletion from the input graph $G$ results in a graph without any induced copy of $H$. $H$-free Edge Completion and $H$-free Edge Editing are defined similarly where only completion (addition) of edges are allowed in the former and both completion and deletion are allowed in the latter. We completely settle the classical complexities of these problems by proving that $H$-free Edge Deletion is NP-complete if and only if $H$ is a graph with at least two edges, $H$-free Edge Completion is NP-comple…
Exact Algorithms for Edge Deletion to Cactus Graphs and Weighted Variants
针对删除最少边使图变为连通仙人掌的问题，提出O*(2^n)时间复杂度的精确算法，并推广到有限不同代价的加权情形。
In the Cluster 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 whose removal from $G$ results a graph in which every connected component is a clique. In this paper we give an algorithm for Cluster Deletion whose running time is $O^*(1.404^k)$.
Exploring Minimal Blocking Sets for Kernelization Lower Bounds for $\mathcal{F}$-Minor Deletion
本研究推广了顶点覆盖问题的极小阻塞集与内核化下界的关联，针对$\boldsymbol{\textit{F}}$-子式删除问题提出新判据，证明了Pw-$\boldsymbol{\textit{η}}$删除等问题无多项式内核，完善了内核化下界的相关结论。
We study the parameterized complexity of the $s$-Club Cluster Edge Deletion problem: Given a graph $G$ and two integers $s \ge 2$ and $k \ge 1$, is it possible to remove at most $k$ edges from $G$ such that each connected component of the resulting graph has diameter at most $s$? This problem is known to be NP-hard already when $s = 2$. We prove that it admits a fixed-parameter tractable algorithm when parameterized by $s$ and the treewidth of the input graph.
In the F-minor-free deletion problem we want to find a minimum vertex set in a given graph that intersects all minor models of graphs from the family F. The Vertex planarization problem is a special case of F-minor-free deletion for the family F = {K_5, K_{3,3}}. Whenever the family F contains at least one planar graph, then F-minor-free deletion is known to admit a constant-factor approximation algorithm and a polynomial kernelization [Fomin, Lokshtanov, Misra, and Saurabh, FOCS'12]. The Vertex planarization problem is arguably the simplest setting for which F does not contain a planar graph …
Dynamic framework for edge-connectivity maintenance of simple graphs
本文提出了一种动态维护简单图k-边连通度的框架，通过稀疏证书和Link-Cut树实现边插入的高效处理，并利用最大流计算恢复边删除后的连通性。
Exact Algorithms for Edge Deletion to Cactus
研究图cactus的边删除问题及生成树到cactus的转换，提出改进的精确算法和多项式时间算法。
We introduce a new dynamic data structure for maintaining the strongly connected components (SCCs) of a directed graph (digraph) under edge deletions, so as to answer a rich repertoire of connectivity queries. Our main technical contribution is a decremental data structure that supports sensitivity queries of the form "are $ u $ and $ v $ strongly connected in the graph $ G \setminus w $?", for any triple of vertices $ u, v, w $, while $ G $ undergoes deletions of edges. Our data structure processes a sequence of edge deletions in a digraph with $n$ vertices in $O(m n \log{n})$ total time and …
We present a dynamic algorithm for maintaining the connected and 2-edge-connected components in an undirected graph subject to edge deletions. The algorithm is Monte-Carlo randomized and processes any sequence of edge deletions in $O(m + n \operatorname{polylog} n)$ total time. Interspersed with the deletions, it can answer queries to whether any two given vertices currently belong to the same (2-edge-)connected component in constant time. Our result is based on a general Monte-Carlo randomized reduction from decremental $c$-edge-connectivity to a variant of fully-dynamic $c$-edge-connectivity…
On the Complexity of the Eigenvalue Deletion Problem
本文系统研究 r-EVD 及其边修改变体的计算复杂性，给出二分图、正则图、簇图等情形下的 NP 完全性、FPT 算法与二次核结果。
We consider the $Π$-free Deletion problem parameterized by the size of a vertex cover, for a range of graph properties $Π$. Given an input graph $G$, this problem asks whether there is a subset of at most $k$ vertices whose removal ensures the resulting graph does not contain a graph from $Π$ as induced subgraph. Many vertex-deletion problems such as Perfect Deletion, Wheel-free Deletion, and Interval Deletion fit into this framework. We introduce the concept of characterizing a graph property $Π$ by low-rank adjacencies, and use it as the cornerstone of a general kernelization theorem for $Π$…
Maintaining Routing Structures under Deletions via Self-Pruning
本文研究在边删除序列下同时维护扩展图路由与修剪的问题，提出一族像超立方体一样易于路由且具有自修剪性的扩展图，并实现最坏情况下每次对抗性删除仅引起少量额外删除，同时可对路由路径长度进行常数因子控制。
We present a decremental data structure for maintaining the SPQR-tree of a planar graph subject to edge contractions and deletions. The update time, amortized over $Ω(n)$ operations, is $O(\log^2 n)$. Via SPQR-trees, we give a decremental data structure for maintaining $3$-vertex connectivity in planar graphs. It answers queries in $O(1)$ time and processes edge deletions and contractions in $O(\log^2 n)$ amortized time. This is an exponential improvement over the previous best bound of $O(\sqrt{n}\,)$ that has stood for over 20 years. In addition, the previous data structures only supported e…
We investigate polynomial-time preprocessing for the problem of hitting forbidden minors in a graph, using the framework of kernelization. For a fixed finite set of connected graphs F, the F-Deletion problem is the following: given a graph G and integer k, is it possible to delete k vertices from G to ensure the resulting graph does not contain any graph from F as a minor? Earlier work by Fomin, Lokshtanov, Misra, and Saurabh [FOCS'12] showed that when F contains a planar graph, an instance (G,k) can be reduced in polynomial time to an equivalent one of size $k^{O(1)}$. In this work we focus o…
We study the parameterized and classical complexity of two related problems on undirected graphs $G=(V,E)$. In Strong Triadic Closure we aim to label the edges in $E$ as strong and weak such that at most~$k$ edges are weak and $G$ contains no induced $P_3$ with two strong edges. In Cluster Deletion, we aim to destroy all induced $P_3$s by a minimum number of edge deletions. We first show that Strong Triadic Closure admits a $4k$-vertex kernel. Then, we study parameterization by $\ell:=|E|-k$ and show that both problems are fixed-parameter tractable and unlikely to admit a polynomial kernel wit…
A 2-club is a graph of diameter at most two. In the decision version of the parametrized {\sc 2-Club Cluster Edge Deletion} problem, an undirected graph $G$ is given along with an integer $k\geq 0$ 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 $\mathcal{O}^*(3^k)$, and a decade-old algorithm was claimed to have an improved running time of $\mathcal{O}^*(2.74^k)$ via a sophisticated case analysis. Unfortunately, this latter algorithm suffers from a flawed b…
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