In Directed Geodetic Set, we are given a (directed) graph and seek a small solution set S ⊆ V(G) such that every vertex lies on a shortest directed path between two vertices in S. While most prior work on Directed Geodetic Set has focused on undirected graphs, in this article we study the problem on directed graphs from the perspective of parameterized complexity. It is known that the problem is W[2]-hard when parameterized by the solution size k, even on directed acyclic graphs (DAGs). We investigate structural parameterizations of the problem. Our first result is a kernel of size 2^O(vcn) for Directed Geodetic Set on general digraphs, where vcn denotes the vertex cover number of the underlying (undirected) graph. This implies an algorithm running in time 2^O(vcn²) ⋅ n^O(1). Furthermore, we prove that, assuming the ETH, the problem does not admit an algorithm running in time 2^o(vcn²) ⋅ n^O(1). Such a tight quadratic exponential lower bound in the parameter is relatively uncommon in parameterized complexity. These results generalize earlier work on undirected graphs by Foucaud et al. [STACS 2025], and complements a recent result on directed graph by Foucaud et al. [CALDAM 2026], that showed that the problem is para-NP-hard for the pathwidth and feedback vertex set number of the underlying graph. Next, we show that on general digraphs, Directed Geodetic Set admits a natural kernel of size (kΔ)^O(rdiam), where Δ is the maximum degree and rdiam denotes the reachability diameter of the digraph (a natural analogue of diameter of undirected graphs). This yields an algorithm running in time (kΔ)^O(rdiam⋅k) ⋅ n^O(1). We further prove that, assuming the ETH, the problem does not admit an algorithm running in time (kΔ)^o(rdiam ⋅ k) ⋅ n^O(1). Finally, we justify the necessity of combining parameters by establishing the following hardness results for Directed Geodetic Set: 1) It is W[2]-hard parameterized by k, even on digraphs of maximum degree 3. 2) It is para-NP-hard parameterized by maximum degree and reachability diameter. One can infer that the problem remains W[2]-hard when parameterized by k, even on graphs of reachability diameter 3 from Araújo and Arraes [DAM 2022]. All our conditional lower bounds and hardness results hold even when the input digraph is restricted to be a DAG.

In this article, we revisit the complexity of the reconfiguration of independent sets under the token sliding rule on chordal graphs. In the \textsc{Token Sliding-Connectivity} problem, the input is a graph $G$ and an integer $k$, and the objective is to determine whether the reconfiguration graph $TS_k(G)$ of $G$ is connected. The vertices of $TS_k(G)$ are $k$-independent sets of $G$, and two vertices are adjacent if and only if one can transform one of the two corresponding independent sets into the other by sliding a vertex (also called a \emph{token}) along an edge. Bonamy and Bousquet [WG'17] proved that the \textsc{Token Sliding-Connectivity} problem is polynomial-time solvable on interval graphs but \NP-hard on split graphs. In light of these two results, the authors asked: can we decide the connectivity of $TS_k(G)$ in polynomial time for chordal graphs with \emph{maximum clique-tree degree} $d$? We answer this question in the negative and prove that the problem is \para-\NP-hard when parameterized by $d$. More precisely, the problem is \NP-hard even when $d = 4$. We then study the parameterized complexity of the problem for a larger parameter called \emph{leafage} and prove that the problem is \co-\W[1]-hard. We prove similar results for a closely related problem called \textsc{Token Sliding-Reachability}. In this problem, the input is a graph $G$ with two of its $k$-independent sets $I$ and $J$, and the objective is to determine whether there is a sequence of valid token sliding moves that transform $I$ into $J$.

In the Isometric Path Partition problem, the input is a graph G with n vertices and an integer k, and the objective is to determine whether the vertices of G can be partitioned into k vertex-disjoint shortest paths. We investigate the parameterized complexity of the problem when parameterized by the treewidth (tw) of the input graph, arguably one of the most widely studied parameters. Courcelle’s theorem [Information & Computation, 1990] shows that graph problems that are expressible as MSO formulas of constant size admit FPT algorithms parameterized by the treewidth of the input graph. This encompasses many natural graph problems. However, many metric-based graph problems, where the solution is defined using some metric-based property of the graph (often the distance) are not expressible as MSO formulas of constant size. These types of problems, Isometric Path Partition being one of them, require individual attention and often draw the boundary for the success story of parameterization by treewidth. We show that Isometric Path Partition is W[1]-hard when parameterized by treewidth (in fact, even pathwidth (pw)), answering the question by Dumas et al. [SIDMA, 2024], Fernau et al. [TCS, 2025], and confirming the aforementioned tendency. We complement this hardness result by designing a tailored dynamic programming algorithm running in n^{O(tw)} time. This dynamic programming approach also results in an algorithm running in time diam^{O(tw²)} ⋅ n^{O(1)}, where diam is the diameter of the graph. It is known that Isometric Path Partition remains NP-hard on graphs of diameter 2; hence, the combination of both parameters is necessary to obtain a tractable algorithm. Note that the dependency on treewidth is unusually high, as most problems that are FPT for treewidth admit algorithms running in time 2^{O(tw)}⋅ n^{O(1)} or 2^{O(tw log (tw))}⋅ n^{O(1)}. However, we rule out the possibility of a significantly faster algorithm, showing that Isometric Path Partition does not admit an algorithm running in time diam^{o(pw²/(log³(pw)))} ⋅ n^{O(1)}, assuming the Randomized-ETH.

We study the \textsc{Labeled Contractibility} problem, where the input consists of two vertex-labeled graphs $G$ and $H$, and the goal is to determine whether $H$ can be obtained from $G$ via a sequence of edge contractions. Lafond and Marchand~[WADS 2025] initiated the parameterized complexity study of this problem, showing it to be \(\W[1]\)-hard when parameterized by the number \(k\) of allowed contractions. They also proved that the problem is fixed-parameter tractable when parameterized by the tree-width \(\tw\) of \(G\), via an application of Courcelle's theorem resulting in a non-constructive algorithm. In this work, we present a constructive fixed-parameter algorithm for \textsc{Labeled Contractibility} with running time \(2^{\mathcal{O}(\tw^2)} \cdot |V(G)|^{\mathcal{O}(1)}\). We also prove that unless the Exponential Time Hypothesis (Ð) fails, it does not admit an algorithm running in time \(2^{o(\tw^2)} \cdot |V(G)|^{\mathcal{O}(1)}\). This result adds \textsc{Labeled Contractibility} to a small list of problems that admit such a lower bound and matching algorithm. We further strengthen existing hardness results by showing that the problem remains \NP-complete even when both input graphs have bounded maximum degree. We also investigate parameterizations by \((k + δ(G))\) where \(δ(G)\) denotes the degeneracy of \(G\), and rule out the existence of subexponential-time algorithms. This answers question raised in Lafond and Marchand~[WADS 2025]. We additionally provide an improved \FPT\ algorithm with better dependence on \((k + δ(G))\) than previously known. Finally, we analyze a brute-force algorithm for \textsc{Labeled Contractibility} with running time \(|V(H)|^{\mathcal{O}(|V(G)|)}\), and show that this running time is optimal under Ð.

We prove a robust contraction decomposition theorem for $H$-minor-free graphs, which states that given an $H$-minor-free graph $G$ and an integer $p$, one can partition in polynomial time the vertices of $G$ into $p$ sets $Z_1,\dots,Z_p$ such that $\operatorname{tw}(G/(Z_i \setminus Z')) = O(p + |Z'|)$ for all $i \in [p]$ and $Z' \subseteq Z_i$. Here, $\operatorname{tw}(\cdot)$ denotes the treewidth of a graph and $G/(Z_i \setminus Z')$ denotes the graph obtained from $G$ by contracting all edges with both endpoints in $Z_i \setminus Z'$. Our result generalizes earlier results by Klein [SICOMP 2008] and Demaine et al. [STOC 2011] based on partitioning $E(G)$, and some recent theorems for planar graphs by Marx et al. [SODA 2022], for bounded-genus graphs (more generally, almost-embeddable graphs) by Bandyapadhyay et al. [SODA 2022], and for unit-disk graphs by Bandyapadhyay et al. [SoCG 2022]. The robust contraction decomposition theorem directly results in parameterized algorithms with running time $2^{\widetilde{O}(\sqrt{k})} \cdot n^{O(1)}$ or $n^{O(\sqrt{k})}$ for every vertex/edge deletion problems on $H$-minor-free graphs that can be formulated as Permutation CSP Deletion or 2-Conn Permutation CSP Deletion. Consequently, we obtain the first subexponential-time parameterized algorithms for Subset Feedback Vertex Set, Subset Odd Cycle Transversal, Subset Group Feedback Vertex Set, 2-Conn Component Order Connectivity on $H$-minor-free graphs. For other problems which already have subexponential-time parameterized algorithms on $H$-minor-free graphs (e.g., Odd Cycle Transversal, Vertex Multiway Cut, Vertex Multicut, etc.), our theorem gives much simpler algorithms of the same running time.

In the Geodetic Set problem, the input consists of a graph G and a positive integer k. The goal is to determine whether there exists a subset S of vertices of size k such that every vertex in the graph is included in a shortest path between two vertices in S. Kellerhals and Koana [IPEC 2020; J. Graph Algorithms Appl 2022] proved that the problem is W[1]-hard when parameterized by the pathwidth or the feedback vertex set number of the input graph. They posed the question of whether the problem admits an XP-algorithm when parameterized by the combination of these two parameters. We answer this in the negative by proving that the problem remains NP-hard even on graphs of constant pathwidth and feedback vertex set number.

The VC-dimension is a well-studied and fundamental complexity measure of a set system (or hypergraph) that is central to many areas of machine learning. We establish several new results on the complexity of computing the VC-dimension. In particular, given a hypergraph $\mathcal{H}=(\mathcal{V},\mathcal{E})$, we prove that the naive $2^{\mathcal{O}(|\mathcal{V}|)}$-time algorithm is asymptotically tight under the Exponential Time Hypothesis (ETH). We then prove that the problem admits a $1$-additive fixed-parameter approximation algorithm when parameterized by the maximum degree of $\mathcal{H}$ and a fixed-parameter algorithm when parameterized by its dimension, and that these are essentially the only such exploitable structural parameters. Lastly, we consider a generalization of the problem, formulated using graphs, which captures the VC-dimension of both set systems and graphs. We design a $2^{\mathcal{O}(\texttt{tw}\cdot \log \texttt{tw})}\cdot |V|$-time algorithm for any graph $G=(V,E)$ of treewidth $\texttt{tw}$ (which, for a set system, applies to the treewidth of its incidence graph). This is in contrast with closely related problems that require a double-exponential dependency on the treewidth (assuming the ETH).

For a graph G, a subset S ⊆ V(G) is called a resolving set of G if, for any two vertices u,v ∈ V(G), there exists a vertex w ∈ S such that d(w,u) ≠ d(w,v). The Metric Dimension problem takes as input a graph G on n vertices and a positive integer k, and asks whether there exists a resolving set of size at most k. In another metric-based graph problem, Geodetic Set, the input is a graph G and an integer k, and the objective is to determine whether there exists a subset S ⊆ V(G) of size at most k such that, for any vertex u ∈ V(G), there are two vertices s₁, s₂ ∈ S such that u lies on a shortest path from s₁ to s₂. These two classical problems are known to be intractable with respect to the natural parameter, i.e., the solution size, as well as most structural parameters, including the feedback vertex set number and pathwidth. We observe that both problems admit an FPT algorithm running in 2^𝒪(vc²) ⋅ n^𝒪(1) time, and a kernelization algorithm that outputs a kernel with 2^𝒪(vc) vertices, where vc is the vertex cover number. We prove that unless the Exponential Time Hypothesis (ETH) fails, Metric Dimension and Geodetic Set, even on graphs of bounded diameter, do not admit - an FPT algorithm running in 2^o(vc²) ⋅ n^𝒪(1) time, nor - a kernelization algorithm that does not increase the solution size and outputs a kernel with 2^o(vc) vertices. We only know of one other problem in the literature that admits such a tight algorithmic lower bound with respect to vc. Similarly, the list of known problems with exponential lower bounds on the number of vertices in kernelized instances is very short.

Treewidth (tw) is an important parameter that, when bounded, yields tractability for many problems. For example, graph problems expressible in Monadic Second Order (MSO) logic and QUANTIFIED SAT or, more generally, QUANTIFIED CSP, are FPT parameterized by the tw of the input's (primal) graph plus the length of the MSO-formula [Courcelle, Information & Computation 1990] and the quantifier rank [Chen, ECAI 2004], resp. The algorithms from these (meta-)results have running times whose dependence on tw is a tower of exponents. A conditional lower bound by Fichte et al. [LICS 2020] shows that, for QUANTIFIED SAT, the height of this tower is equal to the number of quantifier alternations. Lower bounds showing that at least double-exponential factors in the running time are necessary are rare: there are very few (for tw and vertex cover vc parameterizations) and they are for problems that are complete for #NP, $Σ_2^p$, $Π_2^p$, or higher levels of the polynomial hierarchy. We show, for the first time, that it is not necessary to go higher up in the polynomial hierarchy to obtain such lower bounds. We design a novel, yet simple versatile technique based on Sperner families to obtain such lower bounds and apply it to 3 problems: METRIC DIMENSION, STRONG METRIC DIMENSION, and GEODETIC SET. We prove that they do not admit $2^{2^{o(tw)}} \cdot n^{O(1)}$-time algorithms, even on bounded diameter graphs, unless the ETH fails. For STRONG METRIC DIMENSION, the lower bound holds even for vc. We complement our lower bounds with matching upper bounds.

Abstract The leafage of a chordal graph G is the minimum integer $$\ell $$ ℓ such that G can be realized as an intersection graph of subtrees of a tree with $$\ell $$ ℓ leaves. We consider structural parameterization by the leafage of classical domination and cut problems on chordal graphs. Fomin, Golovach, and Raymond [ESA 2018, Algorithmica 2020] proved, among other things, that Dominating Set on chordal graphs admits an algorithm running in time $$2^{\mathcal {O}(\ell ^2)} \cdot n^{\mathcal {O}(1)}$$ 2 O ( ℓ 2 ) · n O ( 1 ) . We present a conceptually much simpler algorithm that runs in time $$2^{\mathcal {O}(\ell )} \cdot n^{\mathcal {O}(1)}$$ 2 O ( ℓ ) · n O ( 1 ) . We extend our approach to obtain similar results for Connected Dominating Set and Steiner Tree. We then consider the two classical cut problems MultiCut with Undeletable Terminals and Multiway Cut with Undeletable Terminals. We prove that the former is [1]-hard when parameterized by the leafage and complement this result by presenting a simple $$n^{\mathcal {O}(\ell )}$$ n O ( ℓ ) -time algorithm. To our surprise, we find that Multiway Cut with Undeletable Terminals on chordal graphs can be solved, in contrast, in $$n^{{{\mathcal {O}}}(1)}$$ n O ( 1 ) -time.

Given a graph G on n vertices and two integers k and d, the Contraction(vc) problem asks whether one can contract at most k edges to reduce the vertex cover number of G by at least d. Recently, Lima et al. [JCSS 2021] proved that Contraction(vc) admits an XP algorithm running in time f(d)⋅nO(d). They asked whether this problem is FPT under this parameterization. In this article, we prove that: (i) Contraction(vc) is W[1]-hard parameterized by k+d. Moreover, unless the ETH fails, the problem does not admit an algorithm running in time f(k+d)⋅no(k+d) for any function f. This answers negatively the open question stated in Lima et al. [JCSS 2021]. (ii) Contraction(vc) is NP-hard even when k=d. (iii) Contraction(vc) can be solved in time 2O(d)⋅nk−d+O(1). This improves the algorithm of Lima et al. [JCSS 2021], and shows that when k=d, Contraction(vc) is FPT parameterized by d (or by k).

Abstract. For a graph [Formula: see text], a subset [Formula: see text] is called a resolving set if for any two vertices [Formula: see text], there exists a vertex [Formula: see text] such that [Formula: see text]. The Metric Dimension problem takes as input a graph [Formula: see text] and a positive integer [Formula: see text], and asks whether there exists a resolving set of size at most [Formula: see text]. This problem was introduced in the 1970s and is known to be NP -hard [M. R. Garey and D. S. Johnson, Computers and Intractability—A Guide to NP-Completeness, Freeman, San Francisco, 1979]. In the realm of parameterized complexity, Hartung and Nichterlein [28 th Conference on Computational Complexity, IEEE, Piscataway, NJ, 2013, pp. 266–276] proved that the problem is W [2]-hard when parameterized by the natural parameter [Formula: see text]. They also observed that it is fixed parameter tractable ( FPT) when parameterized by the vertex cover number and asked about its complexity under smaller parameters, in particular, the feedback vertex set number. We answer this question by proving that Metric Dimension is W [1]-hard when parameterized by the combined parameter feedback vertex set number plus pathwidth. This also improves the result of Bonnet and Purohit [IPEC 2019] which states that the problem is W [1]-hard parameterized by the pathwidth. On the positive side, we show that Metric Dimension is FPT when parameterized by either the distance to cluster or the distance to cocluster, both of which are smaller parameters than the vertex cover number.

The game of rendezvous with adversaries is a game on a graph played by two players: Facilitator and Divider. Facilitator has two agents and Divider has a team of $k \ge 1$ agents. While the initial positions of Facilitator's agents are fixed, Divider gets to select the initial positions of his agents. Then, they take turns to move their agents to adjacent vertices (or stay put) with Facilitator's goal to bring both her agents at same vertex and Divider's goal to prevent it. The computational question of interest is to determine if Facilitator has a winning strategy against Divider with $k$ agents. Fomin, Golovach, and Thilikos [WG, 2021] introduced this game and proved that it is PSPACE-hard and co-W[2]-hard parameterized by the number of agents. This hardness naturally motivates the structural parameterization of the problem. The authors proved that it admits an FPT algorithm when parameterized by the modular width and the number of allowed rounds. However, they left open the complexity of the problem from the perspective of other structural parameters. In particular, they explicitly asked whether the problem admits an FPT or XP-algorithm with respect to the treewidth of the input graph. We answer this question in the negative and show that Rendezvous is co-NP-hard even for graphs of constant treewidth. Further, we show that the problem is co-W[1]-hard when parameterized by the feedback vertex set number and the number of agents, and is unlikely to admit a polynomial kernel when parameterized by the vertex cover number and the number of agents. Complementing these hardness results, we show that the Rendezvous is FPT when parameterized by both the vertex cover number and the solution size. Finally, for graphs of treewidth at most two and girds, we show that the problem can be solved in polynomial time.

For a graph G, a subset S ⊆ V(G) is called a resolving set if for any two vertices u,v ∈ V(G), there exists a vertex w ∈ S such that d(w,u) ≠ d(w,v). The Metric Dimension problem takes as input a graph G and a positive integer k, and asks whether there exists a resolving set of size at most k. This problem was introduced in the 1970s and is known to be NP-hard [GT 61 in Garey and Johnson’s book]. In the realm of parameterized complexity, Hartung and Nichterlein [CCC 2013] proved that the problem is W[2]-hard when parameterized by the natural parameter k. They also observed that it is FPT when parameterized by the vertex cover number and asked about its complexity under smaller parameters, in particular the feedback vertex set number. We answer this question by proving that Metric Dimension is W[1]-hard when parameterized by the feedback vertex set number. This also improves the result of Bonnet and Purohit [IPEC 2019] which states that the problem is W[1]-hard parameterized by the treewidth. Regarding the parameterization by the vertex cover number, we prove that Metric Dimension does not admit a polynomial kernel under this parameterization unless NP ⊆ coNP/poly. We observe that a similar result holds when the parameter is the distance to clique. On the positive side, we show that Metric Dimension is FPT when parameterized by either the distance to cluster or the distance to co-cluster, both of which are smaller parameters than the vertex cover number.

International audience

Subexponential parameterized algorithms are known for a wide range of natural problems on planar graphs, but the techniques are usually highly problem specific. The goal of this paper is to introduce a framework for obtaining time algorithms for a family of graph modification problems that includes problems that can be seen as generalized cycle hitting problems. Our starting point is the Node Unique Label Cover problem (that is, given a CSP instance where each constraint is a permutation of values on two variables, the task is to delete k variables to make the instance satisfiable). We introduce a variant of the problem where k vertices have to be deleted such that every 2-connected component of the remaining instance is satisfiable. Then we extend the problem with cardinality constraints that restrict the number of times a certain value can be used (globally or within a 2-connected component of the solution). We show that there is an time algorithm on planar graphs for any problem that can be formulated this way, which includes a large number of well-studied problems, for example, Odd Cycle Transversal, Subset Feedback Vertex Set, Group Feedback Vertex Set, Subset Group Feedback Vertex Set, Vertex Multiway Cut, and Component Order Connectivity. For those problems that admit appropriate (quasi)polynomial kernels (that increase the parameter only linearly and preserve planarity), our results immediately imply time parameterized algorithms on planar graphs. In particular, we use or adapt known kernelization results to obtain time (randomized) algorithms for Vertex Multiway Cut, Group Feedback Vertex Set, and Subset Feedback Vertex Set. Our algorithms are designed with possible generalization to H-minor free graphs in mind. To obtain the same time algorithms on H-minor free graphs, the only missing piece is the vertex version of a contraction decomposition theorem that we currently have only for planar graphs.

In the Maximum Degree Contraction problem, input is a graph G on n vertices, and integers k, d, and the objective is to check whether G can be transformed into a graph of maximum degree at most d, using at most k edge contractions. A simple brute-force algorithm that checks all possible sets of edges for a solution runs in time n^𝒪(k). As our first result, we prove that this algorithm is asymptotically optimal, upto constants in the exponents, under Exponential Time Hypothesis (ETH). Belmonte, Golovach, van't Hof, and Paulusma studied the problem in the realm of Parameterized Complexity and proved, among other things, that it admits an FPT algorithm running in time (d + k)^(2k) ⋅ n^𝒪(1) = 2^𝒪(k log (k+d)) ⋅ n^𝒪(1), and remains NP-hard for every constant d ≥ 2 (Acta Informatica (2014)). We present a different FPT algorithm that runs in time 2^𝒪(dk) ⋅ n^𝒪(1). In particular, our algorithm runs in time 2^𝒪(k) ⋅ n^𝒪(1), for every fixed d. In the same article, the authors asked whether the problem admits a polynomial kernel, when parameterized by k + d. We answer this question in the negative and prove that it does not admit a polynomial compression unless NP ⊆ coNP/poly.

A graph $G$ is contractible to a graph $H$ if there is a set $X \subseteq E(G)$, such that $G/X$ is isomorphic to $H$. Here, $G/X$ is the graph obtained from $G$ by contracting all the edges in $X$. For a family of graphs $\cal F$, the $\mathcal{F}$-Contraction problem takes as input a graph $G$ on $n$ vertices, and the objective is to output the largest integer $t$, such that $G$ is contractible to a graph $H \in {\cal F}$, where $|V(H)|=t$. When $\cal F$ is the family of paths, then the corresponding $\mathcal{F}$-Contraction problem is called Path Contraction. The problem Path Contraction admits a simple algorithm running in time $2^{n}\cdot n^{{\mathcal{O}}(1)}$. In spite of the deceptive simplicity of the problem, beating the $2^{n}\cdot n^{{\mathcal{O}}(1)}$ bound for Path Contraction seems quite challenging. In this paper, we design an exact exponential time algorithm for Path Contraction that runs in time ${{1.99987}^n}\cdot n^{{\mathcal O}(1)}$. We also define a problem called 3-Disjoint Connected Subgraphs and design an algorithm for it that runs in time $1.88^n\cdot n^{{\mathcal O}(1)}$. The above algorithm is used as a subroutine in our algorithm for Path Contraction.

A graph G is contractible to a graph H if there is a set X subseteq E(G), such that G/X is isomorphic to H. Here, G/X is the graph obtained from G by contracting all the edges in X. For a family of graphs F, the F-Contraction problem takes as input a graph G on n vertices, and the objective is to output the largest integer t, such that G is contractible to a graph H in F, where |V(H)|=t. When F is the family of paths, then the corresponding F-Contraction problem is called Path Contraction. The problem Path Contraction admits a simple algorithm running in time 2^n * n^{O(1)}. In spite of the deceptive simplicity of the problem, beating the 2^n * n^{O(1)} bound for Path Contraction seems quite challenging. In this paper, we design an exact exponential time algorithm for Path Contraction that runs in time 1.99987^n * n^{O(1)}. We also define a problem called 3-Disjoint Connected Subgraphs, and design an algorithm for it that runs in time 1.88^n * n^{O(1)}. The above algorithm is used as a sub-routine in our algorithm for Path Contraction.

For a family of graphs F, the F-Contraction problem takes as an input a graph G and an integer k, and the goal is to decide if there exists S \subseteq E(G) of size at most k such that G/S belongs to F. Here, G/S is the graph obtained from G by contracting all the edges in S. Heggernes et al.[Algorithmica (2014)] were the first to study edge contraction problems in the realm of Parameterized Complexity. They studied \cal F-Contraction when F is a simple family of graphs such as trees and paths. In this paper, we study the F-Contraction problem, where F generalizes the family of trees. In particular, we define this generalization in a "parameterized way". Let T_\ell be the family of graphs such that each graph in T_\ell can be made into a tree by deleting at most \ell edges. Thus, the problem we study is T_\ell-Contraction. We design an FPT algorithm for T_\ell-Contraction running in time O((\ncol)^{O(k + \ell)} * n^{O(1)}). Furthermore, we show that the problem does not admit a polynomial kernel when parameterized by k. Inspired by the negative result for the kernelization, we design a lossy kernel for T_\ell-Contraction of size O([k(k + 2\ell)] ^{(\lceil {\frac{\alpha}{\alpha-1}\rceil + 1)}}).

We study some well-known graph contraction problems in the recently introduced framework of lossy kernelization. In classical kernelization, given an instance (I,k) of a parameterized problem, we are interested in obtaining (in polynomial time) an equivalent instance (I',k') of the same problem whose size is bounded by a function in k. This notion however has a major limitation. Given an approximate solution to the instance (I',k'), we can say nothing about the original instance (I,k). To handle this issue, among others, the framework of lossy kernelization was introduced. In this framework, for a constant alpha, given an instance (I,k) we obtain an instance (I',k') of the same problem such that, for every c>1, any c-approximate solution to (I',k') can be turned into a (c*alpha)-approximate solution to the original instance (I, k) in polynomial time. Naturally, we are interested in a polynomial time algorithm for this task, and further require that |I'| + k' = k^{O(1)}. Akin to the notion of polynomial time approximation schemes in approximation algorithms, a parameterized problem is said to admit a polynomial size approximate kernelization scheme (PSAKS) if it admits a polynomial size alpha-approximate kernel for every approximation parameter alpha > 1. In this work, we design PSAKSs for Tree Contraction, Star Contraction, Out-Tree Contraction and Cactus Contraction problems. These problems do not admit polynomial kernels, and we show that each of them admit a PSAKS with running time k^{f(alpha)}|I|^{O(1)} that returns an instance of size k^{g(alpha)} where f(alpha) and g(alpha) are constants depending on alpha.

In this work, we study the parameterized complexity of various classical graph-theoretic problems in the dynamic framework where the input graph is being updated by a sequence of edge additions and deletions. Vertex subset problems on graphs typically deal with finding a subset of vertices having certain properties that are of interest to us. In real-world applications, the graph under consideration often changes over time and due to this dynamics, the solution at hand might lose the desired properties. The goal in the area of dynamic graph algorithms is to efficiently maintain a solution under these changes. Recomputing a new solution on the new graph is an expensive task especially when the number of modifications made to the graph is significantly smaller than the size of the graph. In the context of parameterized algorithms, two natural parameters are the size k of the symmetric difference of the edge sets of the two graphs (on n vertices) and the size r of the symmetric difference of the two solutions. We study the Dynamic Pi-Deletion problem which is the dynamic variant of the Pi-Deletion problem and show NP-hardness, fixed-parameter tractability and kernelization results. For specific cases of Dynamic Pi-Deletion such as Dynamic Vertex Cover and Dynamic Feedback Vertex Set, we describe improved FPT algorithms and give linear kernels. Specifically, we show that Dynamic Vertex Cover admits algorithms with running times 1.1740^k*n^{O(1)} (polynomial space) and 1.1277^k*n^{O(1)} (exponential space). Then, we show that Dynamic Feedback Vertex Set admits a randomized algorithm with 1.6667^k*n^{O(1)} running time. Finally, we consider Dynamic Connected Vertex Cover, Dynamic Dominating Set and Dynamic Connected Dominating Set and describe algorithms with 2^k*n^{O(1)} running time improving over the known running time bounds for these problems. Additionally, for Dynamic Dominating Set and Dynamic Connected Dominating Set, we show that this is the optimal running time (up to polynomial factors) assuming the Set Cover Conjecture.