The $k$-leaf power graph $G$ of a tree $T$ is a graph whose vertices are the leaves of $T$ and whose edges connect pairs of leaves at unweighted distance at most~$k$ in $T$. Recognition of the $k$-leaf power graphs for $k \geq 7$ is still an open problem. In this paper, we provide two algorithms for this problem for sparse leaf power graphs. Our results shows that the problem of recognizing these graphs is fixed-parameter tractable when parameterized both by $k$ and by the degeneracy of the given graph. To prove this, we first describe how to embed the leaf root of a leaf power graph into a product of the graph with a cycle graph. We bound the treewidth of the resulting product in terms of $k$ and the degeneracy of $G$. The first presented algorithm uses methods based on monadic second-order logic (MSO$_2$) to recognize the existence of a leaf power as a subgraph of the product graph. Using the same embedding in the product graph, the second algorithm presents a dynamic programming approach to solve the problem and provide a better dependence on the parameters.
The metric dimension of a graph is the size of the smallest set of vertices whose distances distinguish all pairs of vertices in the graph. We show that this graph invariant may be calculated by an algorithm whose running time is linear in the input graph size, added to a function of the largest possible number of leaves in a spanning tree of the graph.
We prove that finding a rooted subtree with at least $k$ leaves in a digraph is a fixed parameter tractable problem. A similar result holds for finding rooted spanning trees with many leaves in digraphs from a wide family $\cal L$ that includes all strong and acyclic digraphs. This settles completely an open question of Fellows and solves another one for digraphs in $\cal L$. Our algorithms are based on the following combinatorial result which can be viewed as a generalization of many results for a `spanning tree with many leaves' in the undirected case, and which is interesting on its own: If…
A $k$-leaf power of a tree $T$ is a graph $G$ whose vertices are the leaves of $T$ and whose edges connect pairs of leaves whose distance in $T$ is at most $k$. A graph is a leaf power if it is a $k$-leaf power for some $k$. Over 20 years ago, Nishimura et al. [J. Algorithms, 2002] asked if recognition of leaf powers was in P. Recently, Lafond [SODA 2022] showed an XP algorithm when parameterized by $k$, while leaving the main question open. In this paper, we explore this question from the perspective of two alternative models of leaf powers, showing that both a linear and a star variant of le…
Given an undirected graph $G$ and a parameter $k$, the $k$-Leaf Spanning Tree ($k$-LST) problem asks if $G$ contains a spanning tree with at least $k$ leaves. This problem has been extensively studied over the past three decades. In 2000, Fellows et al. [FSTTCS'00] explicitly asked whether it admits a klam value of 50. A steady progress towards an affirmative answer continued until 5 years ago, when an algorithm of klam value 37 was discovered. In this paper, we present an $O^*(3.188^k)$-time parameterized algorithm for $k$-LST, which shows that the problem admits a klam value of 39. Our algor…
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, H…
Understanding the evolution of a set of genes or species is a fundamental problem in evolutionary biology. The problem we study here takes as input a set of trees describing {possibly discordant} evolutionary scenarios for a given set of genes or species, and aims at finding a single tree that minimizes the leaf-removal distance to the input trees. This problem is a specific instance of the general consensus/supertree problem, widely used to combine or summarize discordant evolutionary trees. The problem we introduce is specifically tailored to address the case of discrepancies between the inp…
The $k^{th}$-power of a given graph $G=(V,E)$ is obtained from $G$ by adding an edge between every two distinct vertices at a distance at most $k$ in $G$. We call $G$ a $k$-Steiner power if it is an induced subgraph of the $k^{th}$-power of some tree. Our main contribution is a polynomial-time recognition algorithm of $4$-Steiner powers, thereby extending the decade-year-old results of (Lin, Kearney and Jiang, ISAAC'00) for $k=1,2$ and (Chang and Ko, WG'07) for $k=3$. A graph $G$ is termed $k$-leaf power if there is some tree $T$ such that: all vertices in $V(G)$ are leaf-nodes of $T$, and $G$…
Grid graphs, and, more generally, $k\times r$ grid graphs, form one of the most basic classes of geometric graphs. Over the past few decades, a large body of works studied the (in)tractability of various computational problems on grid graphs, which often yield substantially faster algorithms than general graphs. Unfortunately, the recognition of a grid graph is particularly hard -- it was shown to be NP-hard even on trees of pathwidth 3 already in 1987. Yet, in this paper, we provide several positive results in this regard in the framework of parameterized complexity (additionally, we present …
The leafage of a chordal graph G is the minimum integer l such that G can be realized as an intersection graph of subtrees of a tree with 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 $2^{O(l^2)} n^{O(1)}$. We present a conceptually much simpler algorithm that runs in time $2^{O(l)} n^{O(1)}$. We extend our approach to obtain similar results for Connected Do…
The Graph Motif problem was introduced in 2006 in the context of biological networks. It consists of deciding whether or not a multiset of colors occurs in a connected subgraph of a vertex-colored graph. Graph Motif has been mostly analyzed from the standpoint of parameterized complexity. The main parameters which came into consideration were the size of the multiset and the number of colors. Though, in the many applications of Graph Motif, the input graph originates from real-life and has structure. Motivated by this prosaic observation, we systematically study its complexity relatively to gr…
The {\sc Directed Maximum Leaf Out-Branching} problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we improve known parameterized algorithms and combinatorial bounds on the number of leaves in out-branchings. We show that \begin{itemize} \item every strongly connected digraph $D$ of order $n$ with minimum in-degree at least 3 has an out-branching with at least $(n/4)^{1/3}-1$ leaves; \item if a strongly connected digraph $D$ does not contain an out-branching with $k$ leaves, then the pathwidth of its und…
The Maximum Agreement Forest (Maf) problem is a well-studied problem in evolutionary biology, which asks for a largest common subforest of a given collection of phylogenetic trees with identical leaf label-set. However, the previous work about the Maf problem are mainly on two binary phylogenetic trees or two general (i.e., binary and non-binary) phylogenetic trees. In this paper, we study the more general version of the problem: the Maf problem on multiple general phylogenetic trees. We present a parameterized algorithm of running time $O(3^k n^2m)$ and a 3-approximation algorithm for the Maf…
In the $k$-Leaf Out-Branching and $k$-Internal Out-Branching problems we are given a directed graph $D$ with a designated root $r$ and a nonnegative integer $k$. The question is to determine the existence of an outbranching rooted at $r$ that has at least $k$ leaves, or at least $k$ internal vertices, respectively. Both these problems were intensively studied from the points of view of parameterized complexity and kernelization, and in particular for both of them kernels with $O(k^2)$ vertices are known on general graphs. In this work we show that $k$-Leaf Out-Branching admits a kernel with $O…
The {\sc $k$-Leaf Out-Branching} problem is to find an out-branching (i.e. a rooted oriented spanning tree) with at least $k$ leaves in a given digraph. The problem has recently received much attention from the viewpoint of parameterized algorithms {alonLNCS4596,AlonFGKS07fsttcs,BoDo2,KnLaRo}. In this paper we step aside and take a kernelization based approach to the {\sc $k$-Leaf-Out-Branching} problem. We give the first polynomial kernel for {\sc Rooted $k$-Leaf-Out-Branching}, a variant of {\sc $k$-Leaf-Out-Branching} where the root of the tree searched for is also a part of the input. Our …
A graph $G$ is a $(Π_A,Π_B)$-graph if $V(G)$ can be bipartitioned into $A$ and $B$ such that $G[A]$ satisfies property $Π_A$ and $G[B]$ satisfies property $Π_B$. The $(Π_{A},Π_{B})$-Recognition problem is to recognize whether a given graph is a $(Π_A,Π_B)$-graph. There are many $(Π_{A},Π_{B})$-Recognition problems, including the recognition problems for bipartite, split, and unipolar graphs. We present efficient algorithms for many cases of $(Π_A,Π_B)$-Recognition based on a technique which we dub inductive recognition. In particular, we give fixed-parameter algorithms for two NP-hard $(Π_{A},…
Connectivity problems like k-Path and k-Disjoint Paths relate to many important milestones in parameterized complexity, namely the Graph Minors Project, color coding, and the recent development of techniques for obtaining kernelization lower bounds. This work explores the existence of polynomial kernels for various path and cycle problems, by considering nonstandard parameterizations. We show polynomial kernels when the parameters are a given vertex cover, a modulator to a cluster graph, or a (promised) max leaf number. We obtain lower bounds via cross-composition, e.g., for Hamiltonian Cycle …
Vertex integrity is a graph parameter that measures the connectivity of a graph. Informally, its meaning is that a graph has small vertex integrity if it has a small separator whose removal disconnects the graph into connected components which are themselves also small. Graphs with low vertex integrity are extremely structured; this renders many hard problems tractable and has recently attracted interest in this notion from the parameterized complexity community. In this paper we revisit the NP-complete problem of computing the vertex integrity of a given graph from the point of view of struct…
The Minimum Eccentricity Shortest Path Problem consists in finding a shortest path with minimum eccentricity in a given undirected graph. The problem is known to be NP-complete and W[2]-hard with respect to the desired eccentricity. We present fpt-algorithms for the problem parameterized by the modular width, distance to cluster graph, the combination of treewidth with the desired eccentricity, and maximum leaf number.
Phylogenetic trees and networks are leaf-labelled graphs used to model evolution. Display graphs are created by identifying common leaf labels in two or more phylogenetic trees or networks. The treewidth of such graphs is bounded as a function of many common dissimilarity measures between phylogenetic trees and this has been leveraged in fixed parameter tractability results. Here we further elucidate the properties of display graphs and their interaction with treewidth. We show that it is NP-hard to recognize display graphs, but that display graphs of bounded treewidth can be recognized in lin…
We develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs. We exemplify our approaches with two well studied problems. For the first problem, {\sc $k$-Leaf Out-Branching}, which is to find an oriented spanning tree with at least $k$ leaves, we obtain an algorithm solving the problem in time $2^{O(\sqrt{k} \log k)} n+ n^{O(1)}$ on directed graphs whose underlying undirected graph excludes some fixed graph $H$ as a minor. For the special case when the input directed graph is planar, the running time can be improved to $2^{O(\s…
The two weighted graph problems Node Multiway Cut (NMC) and Subset Feedback Vertex Set (SFVS) both ask for a vertex set of minimum total weight, that for NMC disconnects a given set of terminals, and for SFVS intersects all cycles containing a vertex of a given set. We design a meta-algorithm that allows to solve both problems in time $2^{O(rw^3)}\cdot n^{4}$, $2^{O(q^2\log(q))}\cdot n^{4}$, and $n^{O(k^2)}$ where $rw$ is the rank-width, $q$ the $\mathbb{Q}$-rank-width, and $k$ the mim-width of a given decomposition. This answers in the affirmative an open question raised by Jaffke et al. (Alg…
We study the influence of a graph parameter called modular-width on the time complexity for optimally solving well-known polynomial problems such as Maximum Matching, Triangle Counting, and Maximum $s$-$t$ Vertex-Capacitated Flow. The modular-width of a graph depends on its (unique) modular decomposition tree, and can be computed in linear time $O(n+m)$ for graphs with $n$ vertices and $m$ edges. Modular decompositions are an important tool for graph algorithms, e.g., for linear-time recognition of certain graph classes. Throughout, we obtain efficient parameterized algorithms of running times…
For fixed integers $r,\ell \geq 0$, a graph $G$ is called an {\em $(r,\ell)$-graph} if the vertex set $V(G)$ can be partitioned into $r$ independent sets and $\ell$ cliques. The class of $(r, \ell)$ graphs generalizes $r$-colourable graphs (when $\ell =0)$ and hence not surprisingly, determining whether a given graph is an $(r, \ell)$-graph is \NP-hard even when $r \geq 3$ or $\ell \geq 3$ in general graphs. When $r$ and $\ell$ are part of the input, then the recognition problem is NP-hard even if the input graph is a perfect graph (where the {\sc Chromatic Number} problem is solvable in polyn…
For fixed integers $r,\ell \geq 0$, a graph $G$ is called an {\em $(r,\ell)$-graph} if the vertex set $V(G)$ can be partitioned into $r$ independent sets and $\ell$ cliques. This brings us to the following natural parameterized questions: {\sc Vertex $(r,\ell)$-Partization} and {\sc Edge $(r,\ell)$-Partization}. An input to these problems consist of a graph $G$ and a positive integer $k$ and the objective is to decide whether there exists a set $S\subseteq V(G)$ ($S\subseteq E(G)$) such that the deletion of $S$ from $G$ results in an $(r,\ell)$-graph. These problems generalize well studied pro…
We compute a canonical circular-arc representation for a given circular-arc (CA) graph which implies solving the isomorphism and recognition problem for this class. To accomplish this we split the class of CA graphs into uniform and non-uniform ones and employ a generalized version of the argument given by Köbler et al (2013) that has been used to show that the subclass of Helly CA graphs can be canonized in logspace. For uniform CA graphs our approach works in logspace and in addition to that Helly CA graphs are a strict subset of uniform CA graphs. Thus our result is a generalization of the …
Let $\mbox{interval} + k v$, $\mbox{interval} + k e$, and $\mbox{interval} - k e$ denote the classes of graphs that can be obtained from some interval graph by adding $k$ vertices, adding $k$ edges, and deleting $k$ edges, respectively. When $k$ is small, these graph classes are called almost interval graphs. They are well motivated from computational biology, where the data ought to be represented by an interval graph while we can only expect an almost interval graph for the best. For any fixed $k$, we give linear-time algorithms for recognizing all these classes, and in the case of membershi…
Bidimensionality theory is a powerful framework for the development of metaalgorithmic techniques. It was introduced by Demaine et al. as a tool to obtain sub-exponential time parameterized algorithms for problems on H-minor free graphs. Demaine and Hajiaghayi extended the theory to obtain PTASs for bidimensional problems, and subsequently improved these results to EPTASs. Fomin et. al related the theory to the existence of linear kernels for parameterized problems. In this paper we revisit bidimensionality theory from the perspective of approximation algorithms and redesign the framework for …
In this paper, we revisit the split decomposition of graphs and give new combinatorial and algorithmic results for the class of totally decomposable graphs, also known as the distance hereditary graphs, and for two non-trivial subclasses, namely the cographs and the 3-leaf power graphs. Precisely, we give strutural and incremental characterizations, leading to optimal fully-dynamic recognition algorithms for vertex and edge modifications, for each of these classes. These results rely on a new framework to represent the split decomposition, namely the graph-labelled trees, which also captures t…
