Protecting a Graph with Mobile Guards

Mobile guards on the vertices of a graph are used to defend it against attacks on either its vertices or its edges. Various models for this problem have been proposed. In this survey we describe a number of these models with particular attention to the case when the attack sequence is infinitely long and the guards must induce some particular configuration before each attack, such as a dominating set or a vertex cover. Results from the literature concerning the number of guards needed to successfully defend a graph in each of these problems are surveyed.Comment: 29 pages, two figures, surve

Total 2-domination of proper interval graphs

A set of vertices W of a graph G is a total k-dominating set when every vertex of G has at least k neighbors in W. In a recent article, Chiarelli et al. (2019) prove that a total k-dominating set can be computed in O(n3k) time when G is a proper interval graph with n vertices and m edges. In this note we reduce the time complexity to O(m) for k=2.Fil: Soulignac, Francisco Juan. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación; Argentin

Data Reduction for Graph Coloring Problems

This paper studies the kernelization complexity of graph coloring problems with respect to certain structural parameterizations of the input instances. We are interested in how well polynomial-time data reduction can provably shrink instances of coloring problems, in terms of the chosen parameter. It is well known that deciding 3-colorability is already NP-complete, hence parameterizing by the requested number of colors is not fruitful. Instead, we pick up on a research thread initiated by Cai (DAM, 2003) who studied coloring problems parameterized by the modification distance of the input graph to a graph class on which coloring is polynomial-time solvable; for example parameterizing by the number k of vertex-deletions needed to make the graph chordal. We obtain various upper and lower bounds for kernels of such parameterizations of q-Coloring, complementing Cai's study of the time complexity with respect to these parameters. Our results show that the existence of polynomial kernels for q-Coloring parameterized by the vertex-deletion distance to a graph class F is strongly related to the existence of a function f(q) which bounds the number of vertices which are needed to preserve the NO-answer to an instance of q-List-Coloring on F.Comment: Author-accepted manuscript of the article that will appear in the FCT 2011 special issue of Information & Computatio

Study of the Gromov hyperbolicity constant on graphs

The concept of Gromov hyperbolicity grasps the essence of negatively curved spaces like the classical hyperbolic space and Riemannian manifolds of negative sectional curvature. It is remarkable that a simple concept leads to such a rich general theory. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of any geodesic metric space is equivalent to the hyperbolicity of a graph related to it. In this Ph. D. Thesis we characterize the hyperbolicity constant of interval graphs and circular-arc graphs. Likewise, we provide relationships between dominant sets and the hyperbolicity constant. Finally, we study the invariance of the hyperbolicity constant when the graphs are transformed by several operators.Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de MadridPresidente: Domingo de Guzmán Pestana Galván.- Secretaria: Ana Portilla Ferreira.- Vocal: Eva Tourís Loj