179 research outputs found

    Computing the Clique-width of Cactus Graphs

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    Similar to the tree-width (twd), the clique-width (cwd) is an invariant of graphs. A well known relationship between tree-width and clique-width is that cwd(G) ≤ 3 · 2twd(G)−1. It is also known that tree-width of Cactus graphs is 2, therefore the clique-width for those graphs is smaller or equal than 6. In this paper, it is shown that the clique-width of Cactus graphs is smaller or equal to 4 and we present a polynomial time algorithm which computes exactly a 4-expression

    Combinatorial Problems on HH-graphs

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    Bir\'{o}, Hujter, and Tuza introduced the concept of HH-graphs (1992), intersection graphs of connected subgraphs of a subdivision of a graph HH. They naturally generalize many important classes of graphs, e.g., interval graphs and circular-arc graphs. We continue the study of these graph classes by considering coloring, clique, and isomorphism problems on HH-graphs. We show that for any fixed HH containing a certain 3-node, 6-edge multigraph as a minor that the clique problem is APX-hard on HH-graphs and the isomorphism problem is isomorphism-complete. We also provide positive results on HH-graphs. Namely, when HH is a cactus the clique problem can be solved in polynomial time. Also, when a graph GG has a Helly HH-representation, the clique problem can be solved in polynomial time. Finally, we observe that one can use treewidth techniques to show that both the kk-clique and list kk-coloring problems are FPT on HH-graphs. These FPT results apply more generally to treewidth-bounded graph classes where treewidth is bounded by a function of the clique number

    Algorithms and Bounds for Very Strong Rainbow Coloring

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    A well-studied coloring problem is to assign colors to the edges of a graph GG so that, for every pair of vertices, all edges of at least one shortest path between them receive different colors. The minimum number of colors necessary in such a coloring is the strong rainbow connection number (\src(G)) of the graph. When proving upper bounds on \src(G), it is natural to prove that a coloring exists where, for \emph{every} shortest path between every pair of vertices in the graph, all edges of the path receive different colors. Therefore, we introduce and formally define this more restricted edge coloring number, which we call \emph{very strong rainbow connection number} (\vsrc(G)). In this paper, we give upper bounds on \vsrc(G) for several graph classes, some of which are tight. These immediately imply new upper bounds on \src(G) for these classes, showing that the study of \vsrc(G) enables meaningful progress on bounding \src(G). Then we study the complexity of the problem to compute \vsrc(G), particularly for graphs of bounded treewidth, and show this is an interesting problem in its own right. We prove that \vsrc(G) can be computed in polynomial time on cactus graphs; in contrast, this question is still open for \src(G). We also observe that deciding whether \vsrc(G) = k is fixed-parameter tractable in kk and the treewidth of GG. Finally, on general graphs, we prove that there is no polynomial-time algorithm to decide whether \vsrc(G) \leq 3 nor to approximate \vsrc(G) within a factor n1−εn^{1-\varepsilon}, unless P==NP

    Calculo del clique-width en graficas simples de acuerdo a su estructura

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    El cálculo del cliquewidth, un número entero que es un invariante para gráficas, ha sido estudiado de manera activa, ya que existen problemas catalogados como NP-Completos que tienen complejidad baja si su representación en gráficas tiene cliquewidth acotado. De cierta manera este parametro mide la dificultad de descomponer una gráfica en una estructura llamada árbol (por su topología). La importancia de este invariante radica en que si un problema de gráficas puede ser acotado por ella entonces puede ser resuelto en tiempo polinomial según el teorema principal de Courcelle. Por otra parte el cliquewidth tiene una relación directa con el invariante tree-width con la distinción de que el primero es más general que el segundo. Para calcular este tipo de invariantes se han propuesto en la literatura diferentes procedimientos que dividen la gráfica original en subgráficas las cuales determinan la complejidad, por lo que en la investigación aquí reportada se ha utilizado una descomposición particular de una gráfica simple, la cual consiste en descomponer la gráfica en ciclos simples y árboles. Las gráficas que consisten de ciclos simples y árboles se denominan cactus, sobre las cuales hemos demostrado que el clique-width es menor o igual a 4 lo que mejora la cota establecida por la relación entre el clique-width y el invariante treewidth la cual establece que el cwd(G) ≤ 3·2twd(G)−1. De igual manera se han estudiado otro tipo de gráficas denominadas poligonales, formadas por polígonos con mismo número de lados los cuales comparten entre si una única arista; sobre este tipo de gráficas en esta investigación se ha demostrado que el cliquewidth es igual a 5, de igual manera mejorando la cota conocida por la relación de las invariantes mencionadas anteriormente. Finalmente, estudiando el comportamiento de operaciones de union de estas subgráficas se ha propuesto un método de aproximación para el cálculo del cliquewidth de una gráfica simple de manera general. El algoritmo esta basado en el clásico algoritmo de Disjktra que encuentra el camino mas corto entre dos vértices de una gráfica. Del planteamiento de los algoritmos mencionados anteriormente se obtuvo la publicación de tres artículos, en los que se incluye el desarrollo de las demostraciones para el cálculo del clique-width en los diferentes escenarios de estudio.CONACy

    Edge deletion to tree-like graph classes

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    For a fixed property (graph class) Π\Pi, given a graph GG and an integer kk, the Π\Pi-deletion problem consists in deciding if we can turn GG into a graph with the property Π\Pi by deleting at most kk edges of GG. The Π\Pi-deletion problem is known to be NP-hard for most of the well-studied graph classes (such as chordal, interval, bipartite, planar, comparability and permutation graphs, among others), with the notable exception of trees. Motivated by this fact, in this work we study the deletion problem for some classes close to trees. We obtain NP-hardness results for several classes of sparse graphs, for which we prove that deletion is hard even when the input is a bipartite graph. In addition, we give sufficient structural conditions for the graph class Π\Pi for NP-hardness. In the case of deletion to cactus, we show that the problem becomes tractable when the input is chordal, and we give polynomial-time algorithms for quasi-threshold graphs.Comment: 12 pages, no figure

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

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    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
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