Two infinite families of critical clique–Helly graphs

A graph is clique–Helly if every family of pairwise intersecting (maximal) cliques has non-empty total intersection. Dourado, Protti and Szwarcfiter conjectured that every clique–Helly graph contains a vertex whose removal maintains it as a clique–Helly graph. We present here two infinite families of counterexamples to this conjecture.Fil: Alcón, Liliana Graciela. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; ArgentinaFil: Pizaña, Miguel. Universidad Autónoma Metropolitana; MéxicoFil: Ravenna, Gabriela Susana. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; Argentin

Recovery of disrupted airline operations using k-Maximum Matching in graphs

International audienceBy Berge's theorem, finding a maximum matching in a graph relies on the use of augmenting paths. When no further constraint is added, Edmonds' algorithm allows to compute a maximum matching in polynomial time by sequentially augmenting such paths. Motivated by applications in the scheduling of airline operations, we consider a similar problem where only paths of bounded length can be augmented. Precisely, let k ≥ 1 be an odd integer, a graph G and a matching M of G. What is the maximum size of a matching that can be obtained from M by using only augmenting paths of length at most k? We first prove that this problem can be solved in polynomial time for k ≤ 3 in any graph and that it is NP-complete for any fixed k ≥ 5 in the class of planar bipartite graphs of degree at most 3 and arbitrarily large girth. We then prove that this problem is in P, for any k, in several subclasses of trees such as caterpillars or trees with all vertices of degree at least 3 " far appart ". Moreover, this problem can be solved in time O(n) in the class of n-node trees when k and the maximum degree are fixed parameters. Finally, we consider a more constrained problem where only paths of length exactly k can be augmented. We prove that this latter problem becomes NP-complete for any fixed k ≥ 3 and in trees when k is part of the input

Two infinite families of critical clique-Helly graphs

A graph is clique–Helly if every family of pairwise intersecting (maximal) cliques has non-empty total intersection. Dourado, Protti and Szwarcfiter conjectured that every clique–Helly graph contains a vertex whose removal maintains it as a clique–Helly graph. We present here two infinite families of counterexamples to this conjecture.Instituto de Física La Plat

A new approach on locally checkable problems

By providing a new framework, we extend previous results on locally checkable problems in bounded treewidth graphs. As a consequence, we show how to solve, in polynomial time for bounded treewidth graphs, double Roman domination and Grundy domination, among other problems for which no such algorithm was previously known. Moreover, by proving that fixed powers of bounded degree and bounded treewidth graphs are also bounded degree and bounded treewidth graphs, we can enlarge the family of problems that can be solved in polynomial time for these graph classes, including distance coloring problems and distance domination problems (for bounded distances)

Advancements in Enhancing Resilience of Electrical Distribution Systems: A Review on Frameworks, Metrics, and Technological Innovations

This comprehensive review paper explores power system resilience, emphasizing its evolution, comparison with reliability, and conducting a thorough analysis of the definition and characteristics of resilience. The paper presents the resilience frameworks and the application of quantitative power system resilience metrics to assess and quantify resilience. Additionally, it investigates the relevance of complex network theory in the context of power system resilience. An integral part of this review involves examining the incorporation of data-driven techniques in enhancing power system resilience. This includes the role of data-driven methods in enhancing power system resilience and predictive analytics. Further, the paper explores the recent techniques employed for resilience enhancement, which includes planning and operational techniques. Also, a detailed explanation of microgrid (MG) deployment, renewable energy integration, and peer-to-peer (P2P) energy trading in fortifying power systems against disruptions is provided. An analysis of existing research gaps and challenges is discussed for future directions toward improvements in power system resilience. Thus, a comprehensive understanding of power system resilience is provided, which helps in improving the ability of distribution systems to withstand and recover from extreme events and disruptions

Grafos ORTH[h, s, t] /

Intersection graphs are a topic that has attracted great interest from researchers in the area of graph theory since the 1960s. In this thesis we study the intersection graphs of subtrees of a tree. More precisely we will present results on the class of graphs ORTH[h, s, t]. The graphs that belong to this class are those that admit a representation by intersection of subtrees of a host tree, in which the maximum degree of the host tree is h, the maximum degree of any subtree is s, all leaves of the subtree are also leaves of the host tree and two vertices are adjacent in the graph if and only if their corresponding subtrees have at least t nodes in common and at least one of these nodes is a leaf. Results of representability, non-representability and complexity for classes ORTH[h, s, t] with the different values of the parameters h, s or t are present in this work.Grafos de interseção é um assunto que desperta grande interesse de pesquisadores da área de teoria dos grafos desde a década de 60. Nesta tese estudamos os grafos de interseção de subárvores de uma árvore. Mais precisamente vamos apresentar resultados sobre a classe de grafos ORTH[h, s, t]. Os grafos que pertencem a esta classe são aqueles que admitem uma representação por interseção de subárvores de uma árvore hospedeira, na qual o grau máximo da árvore hospedeira é h, o grau máximo de qualquer subárvore é s, todas as folhas das subárvores também são folhas da árvore hospedeira e dois vértices são adjacentes no grafo se e somente se suas subárvores correspondentes possuem no mínimo t nós em comum e ao menos um destes nós é uma folha. Resultados de representabilidade, não-representabilidade e complexidade para classes ORTH[h, s, t] com a variação do valores dos parâmetros h, s ou t estão presentes neste trabalho