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paloma lima

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Santiago Leiras

Universidad de Buenos Aires

kyriaki (korina) pavlidou

Aristotle University of Thessaloniki

Melanie O'Brien

The University of Western Australia

Patricia Naftali

Fonds National de la Recherche Scientifique

King's College London

Cybele C Almeida

Universidade Federal do Rio Grande do Sul

maddalena carli

University of Teramo

Sergio Sánchez Collantes

Universidad de Burgos

Carina Barbosa Gouvêa

Universidade Federal de Pernambuco

Wagner Feloniuk

FURG - Universidade Federal do Rio Grande

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Papers by paloma lima

к вопросу о дарении предметов роскоши как изменчивой составляющей этапов развития общества особенности получения подарков государственными служащими

Федеральное государственное бюджетное образовательное учреждение высшего образования «Костромской государственный университет», 2015

Graph Square Roots of Small Distance from Degree One Graphs

Given a graph class ℋ, the task of the ℋ-Square Root problem is to decide, whether an input graph... more Given a graph class ℋ, the task of the ℋ-Square Root problem is to decide, whether an input graph G has a square root H from ℋ. We are interested in the parameterized complexity of the problem for classes ℋ that are composed by the graphs at vertex deletion distance at most k from graphs of maximum degree at most one, that is, we are looking for a square root H such that there is a modulator S of size k such that H-S is the disjoint union of isolated vertices and disjoint edges. We show that different variants of the problems with constraints on the number of isolated vertices and edges in H-S are FPT when parameterized by k by demonstrating algorithms with running time 2^2^O(k)· n^O(1). We further show that the running time of our algorithms is asymptotically optimal and it is unlikely that the double-exponential dependence on k could be avoided. In particular, we prove that the VC-k Root problem, that asks whether an input graph has a square root with vertex cover of size at most ...

Finding Connected Secluded Subgraphs

Problems related to finding induced subgraphs satisfying given properties form one of the most st... more Problems related to finding induced subgraphs satisfying given properties form one of the most studied areas within graph algorithms. Such problems have given rise to breakthrough results and led to development of new techniques both within the traditional P vs NP dichotomy and within parameterized complexity. The \Pi-Subgraph problem asks whether an input graph contains an induced subgraph on at least k vertices satisfying graph property \Pi. For many applications, it is desirable that the found subgraph has as few connections to the rest of the graph as possible, which gives rise to the Secluded \Pi-Subgraph problem. Here, input k is the size of the desired subgraph, and input t is a limit on the number of neighbors this subgraph has in the rest of the graph. This problem has been studied from a parameterized perspective, and unfortunately it turns out to be W[1]-hard for many graph properties \Pi, even when parameterized by k+t. We show that the situation changes when we are look...

Classifying k-Edge Colouring for H-free Graphs

A graph is H-free if it does not contain an induced subgraph isomorphic to H. For every integer k... more

b-Coloring Parameterized by Clique-Width

We provide a polynomial-time algorithm for b-Coloring on graphs of constant clique-width. This un... more We provide a polynomial-time algorithm for b-Coloring on graphs of constant clique-width. This unifies and extends previously known polynomial-time results on several graph classes, and answers open questions posed by Campos and Silva [Algorithmica, 2018] and Bonomo et al. [Graphs Combin., 2009]. This constitutes the first result concerning structural parameterizations of this problem. We show that the problem is FPT when parameterized by the vertex cover number on general graphs, and on chordal graphs when parameterized by the number of colors. Additionally, we observe that our algorithm for graphs of bounded clique-width can be adapted to solve the Fall Coloring problem within the same runtime bound.

Algorithms for outerplanar graph roots and graph roots of pathwidth at most 2

Deciding whether a given graph has a square root is a classical problem that has been studied ext... more Deciding whether a given graph has a square root is a classical problem that has been studied extensively both from graph theoretic and from algorithmic perspectives. The problem is NP-complete in general, and consequently substantial effort has been dedicated to deciding whether a given graph has a square root that belongs to a particular graph class. There are both polynomial-time solvable and NP-complete cases, depending on the graph class. We contribute with new results in this direction. Given an arbitrary input graph G, we give polynomial-time algorithms to decide whether G has an outerplanar square root, and whether G has a square root that is of pathwidth at most 2.

On the Maximum Number of Edges in Chordal Graphs of Bounded Degree and Matching Number

We determine the maximum number of edges that a chordal graph G can have if its degree, ∆(G), and... more We determine the maximum number of edges that a chordal graph G can have if its degree, ∆(G), and its matching number, ν(G), are bounded. To do so, we show that for every d, ν ∈ N, there exists a chordal graph G with ∆(G) < d and ν(G) < ν whose number of edges matches the upper bound, while having a simple structure: it is a disjoint union of cliques and stars.

Reducing the domination number of graphs via edge contractions

In this paper, we study the following problem: given a connected graph $G$, can we reduce the dom... more

Reducing graph transversals via edge contractions

J. Comput. Syst. Sci., 2021

For a graph invariant π, the Contraction( π ) problem consists of, given a graph G and positive i... more For a graph invariant π, the Contraction( π ) problem consists of, given a graph G and positive integers k , d , deciding whether one can contract k edges of G to obtain a graph in which π has dropped by at least d. Galby et al. [ISAAC 2019, MFCS 2019] studied the case where π is the size of a minimum dominating set. We focus on graph invariants defined as the minimum size of a vertex set that hits all the occurrences of graphs in a collection H according to a fixed containment relation. We prove co-NP -hardness results under some assumptions on the graphs in H , in particular implying that Contraction( π ) is co-NP -hard for fixed k = d = 1 when π is the size of a minimum feedback vertex set or an odd cycle transversal. In sharp contrast, when π is the size of a minimum vertex cover, the problem is in XP parameterized by d.

Goiânia: Cidade Criativa

Esse poster e resultado das primeiras observacoes sobre o cenario da Economia Criativa no municip... more Esse poster e resultado das primeiras observacoes sobre o cenario da Economia Criativa no municipio da cidade de Goiânia, capital do estado de Goias. Apresentaremos os eventos que possuem a caracteristica principal trabalhos criativos com desenvolvimento e divulgacao colaborativa; festivais musicais, cooperativas, pequenos empreendedores autorais e feiras livres. A metodologia para o desenvolvimento utiliza-se da aplicacao de questionarios estruturados e semi estruturados para consumidores, expositores e organizadores desses eventos para analise qualitativa e quantitativa. Os autores base da pesquisa sao Ana Carla Fonseca (2013), John Howkins (2012), bem como o relatorio da UNCTAD sobre a Economia Criativa no Brasil e no Mundo.

Three Problems on Well-Partitioned Chordal Graphs

Well-partitioned chordal graphs: obstruction set and disjoint paths

We introduce a new subclass of chordal graphs that generalizes split graphs, which we call well-p... more We introduce a new subclass of chordal graphs that generalizes split graphs, which we call well-partitioned chordal graphs. Split graphs are graphs that admit a partition of the vertex set into cliques that can be arranged in a star structure, the leaves of which are of size one. Well-partitioned chordal graphs are a generalization of this concept in the following two ways. First, the cliques in the partition can be arranged in a tree structure, and second, each clique is of arbitrary size. We provide a characterization of well-partitioned chordal graphs by forbidden induced subgraphs, and give a polynomial-time algorithm that given any graph, either finds an obstruction, or outputs a partition of its vertex set that asserts that the graph is well-partitioned chordal. We demonstrate the algorithmic use of this graph class by showing that two variants of the problem of finding pairwise disjoint paths between k given pairs of vertices is in FPT parameterized by k on well-partitioned c...

Blocking dominating sets for $H$-free graphs via edge contractions

In this paper, we consider the following problem: given a connected graph $G$, can we reduce the ... more In this paper, we consider the following problem: given a connected graph $G$, can we reduce the domination number of $G$ by one by using only one edge contraction? We show that the problem is $\mathsf{NP}$-hard when restricted to $\{P_6,P_4+P_2\}$-free graphs and that it is $\mathsf{coNP}$-hard when restricted to subcubic claw-free graphs and $2P_3$-free graphs. As a consequence, we are able to establish a complexity dichotomy for the problem on $H$-free graphs when $H$ is connected.

On the Parameterized Complexity of k-Edge Colouring

ArXiv, 2019

For every fixed integer $k \geq 1$, we prove that $k$-Edge Colouring is fixed-parameter-tractable... more

Algorithms for Outerplanar Graph Roots and Graph Roots of Pathwidth at Most 2

ArXiv, 2017

Deciding whether a given graph has a square root is a classical problem that has been studied ext... more Deciding whether a given graph has a square root is a classical problem that has been studied extensively both from graph theoretic and from algorithmic perspectives. The problem is NP-complete in general, and consequently substantial effort has been dedicated to deciding whether a given graph has a square root that belongs to a particular graph class. There are both polynomial-time solvable and NP-complete cases, depending on the graph class. We contribute with new results in this direction. Given an arbitrary input graph G, we give polynomial-time algorithms to decide whether G has an outerplanar square root, and whether G has a square root that is of pathwidth at most 2.

b-Coloring Parameterized by Clique-Width

ArXiv, 2021

We provide a polynomial-time algorithm for b-Coloring on graphs of constant clique-width. This un... more We provide a polynomial-time algorithm for b-Coloring on graphs of constant clique-width. This unifies and extends previously known polynomial-time results on several graph classes, and answers open questions posed by Campos and Silva [Algorithmica, 2018] and Bonomo et al. [Graphs Combin., 2009]. This constitutes the first result concerning structural parameterizations of this problem. We show that the problem is FPT when parameterized by the vertex cover number on general graphs, and on chordal graphs when parameterized by the number of colors. Additionally, we observe that our algorithm for graphs of bounded clique-width can be adapted to solve the Fall Coloring problem within the same runtime bound.

Finding Connected Secluded Subgraphs

ArXiv, 2017

Problems related to finding induced subgraphs satisfying given properties form one of the most st... more Problems related to finding induced subgraphs satisfying given properties form one of the most studied areas within graph algorithms. Such problems have given rise to breakthrough results and led to development of new techniques both within the traditional P vs NP dichotomy and within parameterized complexity. The \Pi-Subgraph problem asks whether an input graph contains an induced subgraph on at least k vertices satisfying graph property \Pi. For many applications, it is desirable that the found subgraph has as few connections to the rest of the graph as possible, which gives rise to the Secluded \Pi-Subgraph problem. Here, input k is the size of the desired subgraph, and input t is a limit on the number of neighbors this subgraph has in the rest of the graph. This problem has been studied from a parameterized perspective, and unfortunately it turns out to be W[1]-hard for many graph properties \Pi, even when parameterized by k+t. We show that the situation changes when we are look...

Graph Square Roots of Small Distance from Degree One Graphs

Given a graph class $\mathcal{H}$, the task of the $\mathcal{H}$-Square Root problem is to decide... more Given a graph class $\mathcal{H}$, the task of the $\mathcal{H}$-Square Root problem is to decide, whether an input graph $G$ has a square root $H$ from $\mathcal{H}$. We are interested in the parameterized complexity of the problem for classes $\mathcal{H}$ that are composed by the graphs at vertex deletion distance at most $k$ from graphs of maximum degree at most one, that is, we are looking for a square root $H$ such that there is a modulator $S$ of size $k$ such that $H-S$ is the disjoint union of isolated vertices and disjoint edges. We show that different variants of the problems with constraints on the number of isolated vertices and edges in $H-S$ are FPT when parameterized by $k$ by demonstrating algorithms with running time $2^{2^{O(k)}}\cdot n^{O(1)}$. We further show that the running time of our algorithms is asymptotically optimal and it is unlikely that the double-exponential dependence on $k$ could be avoided. In particular, we prove that the VC-$k$ Root problem, tha...

Rainbow Vertex Coloring Bipartite Graphs and Chordal Graphs

Given a graph with colors on its vertices, a path is called a rainbow vertex path if all its inte... more Given a graph with colors on its vertices, a path is called a rainbow vertex path if all its internal vertices have distinct colors. We say that the graph is rainbow vertex-connected if there is a rainbow vertex path between every pair of its vertices. We study the problem of deciding whether the vertices of a given graph can be colored with at most k colors so that the graph becomes rainbow vertex-connected. Although edge-colorings have been studied extensively under similar constraints, there are significantly fewer results on the vertex variant that we consider. In particular, its complexity on structured graph classes was explicitly posed as an open question. We show that the problem remains NP-complete even on bipartite apex graphs and on split graphs. The former can be seen as a first step in the direction of studying the complexity of rainbow coloring on sparse graphs, an open problem which has attracted attention but limited progress. We also give hardness of approximation r...

D ec 2 01 7 Transversals of longest paths ∗

Let lpt(G) be the minimum cardinality of a set of vertices that intersects all longest paths in a... more Let lpt(G) be the minimum cardinality of a set of vertices that intersects all longest paths in a graph G. Let ω(G) be the size of a maximum clique in G, and tw(G) be the treewidth of G. We prove that lpt(G) ≤ max{1, ω(G)− 2} when G is a connected chordal graph; that lpt(G) = 1 when G is a connected bipartite permutation graph or a connected full substar graph; and that lpt(G) ≤ tw(G) for any connected graph G.

$Figure 7: The two cases in the proof of Lemma 27. The solid lines represent the edges in P. Let H be the subgraph of G induced by the vertices {xj : 1 > y}Uf{yj: j = fi}. As Gis connected and G # H, there exists an edge between H and G — V(H). First suppose that such an edge is between a vertex x; in H and a vertex y; in G— V(H). Then, by Proposition 21, x;, is adjacent to y;. Hence, y,x;, - P is also a path, a contradiction. (See Figure 7(a).) Now suppose that there is an edge between a vertex x; in G — V(H) and a vertex y; in H. Then, by Proposition 21, x; is adjacent to y;,. So Q = P—x;,y;, + x1y;, is also a longest path. As V(Q)\ V(P) = {x1}, we have that w ¢ Q. Observe also that v ¢ Q. Indeed, otherwise Q - vw is a path longer than P. Hence, by Proposition 26, all edges of Q are parallel to vw, which implies that v < x; < Xj,,a contradiction to the way P was chosen. (See Figure 7(b).)$

к вопросу о дарении предметов роскоши как изменчивой составляющей этапов развития общества особенности получения подарков государственными служащими

Федеральное государственное бюджетное образовательное учреждение высшего образования «Костромской государственный университет», 2015

Graph Square Roots of Small Distance from Degree One Graphs

Given a graph class ℋ, the task of the ℋ-Square Root problem is to decide, whether an input graph... more Given a graph class ℋ, the task of the ℋ-Square Root problem is to decide, whether an input graph G has a square root H from ℋ. We are interested in the parameterized complexity of the problem for classes ℋ that are composed by the graphs at vertex deletion distance at most k from graphs of maximum degree at most one, that is, we are looking for a square root H such that there is a modulator S of size k such that H-S is the disjoint union of isolated vertices and disjoint edges. We show that different variants of the problems with constraints on the number of isolated vertices and edges in H-S are FPT when parameterized by k by demonstrating algorithms with running time 2^2^O(k)· n^O(1). We further show that the running time of our algorithms is asymptotically optimal and it is unlikely that the double-exponential dependence on k could be avoided. In particular, we prove that the VC-k Root problem, that asks whether an input graph has a square root with vertex cover of size at most ...

Finding Connected Secluded Subgraphs

Problems related to finding induced subgraphs satisfying given properties form one of the most st... more Problems related to finding induced subgraphs satisfying given properties form one of the most studied areas within graph algorithms. Such problems have given rise to breakthrough results and led to development of new techniques both within the traditional P vs NP dichotomy and within parameterized complexity. The \Pi-Subgraph problem asks whether an input graph contains an induced subgraph on at least k vertices satisfying graph property \Pi. For many applications, it is desirable that the found subgraph has as few connections to the rest of the graph as possible, which gives rise to the Secluded \Pi-Subgraph problem. Here, input k is the size of the desired subgraph, and input t is a limit on the number of neighbors this subgraph has in the rest of the graph. This problem has been studied from a parameterized perspective, and unfortunately it turns out to be W[1]-hard for many graph properties \Pi, even when parameterized by k+t. We show that the situation changes when we are look...

Classifying k-Edge Colouring for H-free Graphs

A graph is H-free if it does not contain an induced subgraph isomorphic to H. For every integer k... more

b-Coloring Parameterized by Clique-Width

We provide a polynomial-time algorithm for b-Coloring on graphs of constant clique-width. This un... more We provide a polynomial-time algorithm for b-Coloring on graphs of constant clique-width. This unifies and extends previously known polynomial-time results on several graph classes, and answers open questions posed by Campos and Silva [Algorithmica, 2018] and Bonomo et al. [Graphs Combin., 2009]. This constitutes the first result concerning structural parameterizations of this problem. We show that the problem is FPT when parameterized by the vertex cover number on general graphs, and on chordal graphs when parameterized by the number of colors. Additionally, we observe that our algorithm for graphs of bounded clique-width can be adapted to solve the Fall Coloring problem within the same runtime bound.

Algorithms for outerplanar graph roots and graph roots of pathwidth at most 2

Deciding whether a given graph has a square root is a classical problem that has been studied ext... more Deciding whether a given graph has a square root is a classical problem that has been studied extensively both from graph theoretic and from algorithmic perspectives. The problem is NP-complete in general, and consequently substantial effort has been dedicated to deciding whether a given graph has a square root that belongs to a particular graph class. There are both polynomial-time solvable and NP-complete cases, depending on the graph class. We contribute with new results in this direction. Given an arbitrary input graph G, we give polynomial-time algorithms to decide whether G has an outerplanar square root, and whether G has a square root that is of pathwidth at most 2.

On the Maximum Number of Edges in Chordal Graphs of Bounded Degree and Matching Number

We determine the maximum number of edges that a chordal graph G can have if its degree, ∆(G), and... more We determine the maximum number of edges that a chordal graph G can have if its degree, ∆(G), and its matching number, ν(G), are bounded. To do so, we show that for every d, ν ∈ N, there exists a chordal graph G with ∆(G) < d and ν(G) < ν whose number of edges matches the upper bound, while having a simple structure: it is a disjoint union of cliques and stars.

Reducing the domination number of graphs via edge contractions

In this paper, we study the following problem: given a connected graph $G$, can we reduce the dom... more

Reducing graph transversals via edge contractions

J. Comput. Syst. Sci., 2021

For a graph invariant π, the Contraction( π ) problem consists of, given a graph G and positive i... more For a graph invariant π, the Contraction( π ) problem consists of, given a graph G and positive integers k , d , deciding whether one can contract k edges of G to obtain a graph in which π has dropped by at least d. Galby et al. [ISAAC 2019, MFCS 2019] studied the case where π is the size of a minimum dominating set. We focus on graph invariants defined as the minimum size of a vertex set that hits all the occurrences of graphs in a collection H according to a fixed containment relation. We prove co-NP -hardness results under some assumptions on the graphs in H , in particular implying that Contraction( π ) is co-NP -hard for fixed k = d = 1 when π is the size of a minimum feedback vertex set or an odd cycle transversal. In sharp contrast, when π is the size of a minimum vertex cover, the problem is in XP parameterized by d.

Goiânia: Cidade Criativa

Esse poster e resultado das primeiras observacoes sobre o cenario da Economia Criativa no municip... more Esse poster e resultado das primeiras observacoes sobre o cenario da Economia Criativa no municipio da cidade de Goiânia, capital do estado de Goias. Apresentaremos os eventos que possuem a caracteristica principal trabalhos criativos com desenvolvimento e divulgacao colaborativa; festivais musicais, cooperativas, pequenos empreendedores autorais e feiras livres. A metodologia para o desenvolvimento utiliza-se da aplicacao de questionarios estruturados e semi estruturados para consumidores, expositores e organizadores desses eventos para analise qualitativa e quantitativa. Os autores base da pesquisa sao Ana Carla Fonseca (2013), John Howkins (2012), bem como o relatorio da UNCTAD sobre a Economia Criativa no Brasil e no Mundo.

Three Problems on Well-Partitioned Chordal Graphs

Well-partitioned chordal graphs: obstruction set and disjoint paths

We introduce a new subclass of chordal graphs that generalizes split graphs, which we call well-p... more We introduce a new subclass of chordal graphs that generalizes split graphs, which we call well-partitioned chordal graphs. Split graphs are graphs that admit a partition of the vertex set into cliques that can be arranged in a star structure, the leaves of which are of size one. Well-partitioned chordal graphs are a generalization of this concept in the following two ways. First, the cliques in the partition can be arranged in a tree structure, and second, each clique is of arbitrary size. We provide a characterization of well-partitioned chordal graphs by forbidden induced subgraphs, and give a polynomial-time algorithm that given any graph, either finds an obstruction, or outputs a partition of its vertex set that asserts that the graph is well-partitioned chordal. We demonstrate the algorithmic use of this graph class by showing that two variants of the problem of finding pairwise disjoint paths between k given pairs of vertices is in FPT parameterized by k on well-partitioned c...

Blocking dominating sets for $H$-free graphs via edge contractions

In this paper, we consider the following problem: given a connected graph $G$, can we reduce the ... more In this paper, we consider the following problem: given a connected graph $G$, can we reduce the domination number of $G$ by one by using only one edge contraction? We show that the problem is $\mathsf{NP}$-hard when restricted to $\{P_6,P_4+P_2\}$-free graphs and that it is $\mathsf{coNP}$-hard when restricted to subcubic claw-free graphs and $2P_3$-free graphs. As a consequence, we are able to establish a complexity dichotomy for the problem on $H$-free graphs when $H$ is connected.

On the Parameterized Complexity of k-Edge Colouring

ArXiv, 2019

For every fixed integer $k \geq 1$, we prove that $k$-Edge Colouring is fixed-parameter-tractable... more

Algorithms for Outerplanar Graph Roots and Graph Roots of Pathwidth at Most 2

ArXiv, 2017

Deciding whether a given graph has a square root is a classical problem that has been studied ext... more Deciding whether a given graph has a square root is a classical problem that has been studied extensively both from graph theoretic and from algorithmic perspectives. The problem is NP-complete in general, and consequently substantial effort has been dedicated to deciding whether a given graph has a square root that belongs to a particular graph class. There are both polynomial-time solvable and NP-complete cases, depending on the graph class. We contribute with new results in this direction. Given an arbitrary input graph G, we give polynomial-time algorithms to decide whether G has an outerplanar square root, and whether G has a square root that is of pathwidth at most 2.

b-Coloring Parameterized by Clique-Width

ArXiv, 2021

We provide a polynomial-time algorithm for b-Coloring on graphs of constant clique-width. This un... more We provide a polynomial-time algorithm for b-Coloring on graphs of constant clique-width. This unifies and extends previously known polynomial-time results on several graph classes, and answers open questions posed by Campos and Silva [Algorithmica, 2018] and Bonomo et al. [Graphs Combin., 2009]. This constitutes the first result concerning structural parameterizations of this problem. We show that the problem is FPT when parameterized by the vertex cover number on general graphs, and on chordal graphs when parameterized by the number of colors. Additionally, we observe that our algorithm for graphs of bounded clique-width can be adapted to solve the Fall Coloring problem within the same runtime bound.

Finding Connected Secluded Subgraphs

ArXiv, 2017

Problems related to finding induced subgraphs satisfying given properties form one of the most st... more Problems related to finding induced subgraphs satisfying given properties form one of the most studied areas within graph algorithms. Such problems have given rise to breakthrough results and led to development of new techniques both within the traditional P vs NP dichotomy and within parameterized complexity. The \Pi-Subgraph problem asks whether an input graph contains an induced subgraph on at least k vertices satisfying graph property \Pi. For many applications, it is desirable that the found subgraph has as few connections to the rest of the graph as possible, which gives rise to the Secluded \Pi-Subgraph problem. Here, input k is the size of the desired subgraph, and input t is a limit on the number of neighbors this subgraph has in the rest of the graph. This problem has been studied from a parameterized perspective, and unfortunately it turns out to be W[1]-hard for many graph properties \Pi, even when parameterized by k+t. We show that the situation changes when we are look...

Graph Square Roots of Small Distance from Degree One Graphs

Given a graph class $\mathcal{H}$, the task of the $\mathcal{H}$-Square Root problem is to decide... more Given a graph class $\mathcal{H}$, the task of the $\mathcal{H}$-Square Root problem is to decide, whether an input graph $G$ has a square root $H$ from $\mathcal{H}$. We are interested in the parameterized complexity of the problem for classes $\mathcal{H}$ that are composed by the graphs at vertex deletion distance at most $k$ from graphs of maximum degree at most one, that is, we are looking for a square root $H$ such that there is a modulator $S$ of size $k$ such that $H-S$ is the disjoint union of isolated vertices and disjoint edges. We show that different variants of the problems with constraints on the number of isolated vertices and edges in $H-S$ are FPT when parameterized by $k$ by demonstrating algorithms with running time $2^{2^{O(k)}}\cdot n^{O(1)}$. We further show that the running time of our algorithms is asymptotically optimal and it is unlikely that the double-exponential dependence on $k$ could be avoided. In particular, we prove that the VC-$k$ Root problem, tha...

Rainbow Vertex Coloring Bipartite Graphs and Chordal Graphs

Given a graph with colors on its vertices, a path is called a rainbow vertex path if all its inte... more Given a graph with colors on its vertices, a path is called a rainbow vertex path if all its internal vertices have distinct colors. We say that the graph is rainbow vertex-connected if there is a rainbow vertex path between every pair of its vertices. We study the problem of deciding whether the vertices of a given graph can be colored with at most k colors so that the graph becomes rainbow vertex-connected. Although edge-colorings have been studied extensively under similar constraints, there are significantly fewer results on the vertex variant that we consider. In particular, its complexity on structured graph classes was explicitly posed as an open question. We show that the problem remains NP-complete even on bipartite apex graphs and on split graphs. The former can be seen as a first step in the direction of studying the complexity of rainbow coloring on sparse graphs, an open problem which has attracted attention but limited progress. We also give hardness of approximation r...

D ec 2 01 7 Transversals of longest paths ∗

Let lpt(G) be the minimum cardinality of a set of vertices that intersects all longest paths in a... more Let lpt(G) be the minimum cardinality of a set of vertices that intersects all longest paths in a graph G. Let ω(G) be the size of a maximum clique in G, and tw(G) be the treewidth of G. We prove that lpt(G) ≤ max{1, ω(G)− 2} when G is a connected chordal graph; that lpt(G) = 1 when G is a connected bipartite permutation graph or a connected full substar graph; and that lpt(G) ≤ tw(G) for any connected graph G.

$Figure 7: The two cases in the proof of Lemma 27. The solid lines represent the edges in P. Let H be the subgraph of G induced by the vertices {xj : 1 > y}Uf{yj: j = fi}. As Gis connected and G # H, there exists an edge between H and G — V(H). First suppose that such an edge is between a vertex x; in H and a vertex y; in G— V(H). Then, by Proposition 21, x;, is adjacent to y;. Hence, y,x;, - P is also a path, a contradiction. (See Figure 7(a).) Now suppose that there is an edge between a vertex x; in G — V(H) and a vertex y; in H. Then, by Proposition 21, x; is adjacent to y;,. So Q = P—x;,y;, + x1y;, is also a longest path. As V(Q)\ V(P) = {x1}, we have that w ¢ Q. Observe also that v ¢ Q. Indeed, otherwise Q - vw is a path longer than P. Hence, by Proposition 26, all edges of Q are parallel to vw, which implies that v < x; < Xj,,a contradiction to the way P was chosen. (See Figure 7(b).)$