New algorithms for weighted k-domination and total k-domination problems in proper interval graphs

• Autores: Chiarelli, Nina; Hartinger, Tatiana Romina; Leoni, Valeria Alejandra; Lopez Pujato, María Inés; Milanič, Martin
• Año de publicación: 2019
• Idioma: inglés
• Tipo de recurso: artículo
• Materia: K-DOMINATION; POLYNOMIAL-TIME ALGORITHM; PROPER INTERVAL GRAPH; TOTAL K-DOMINATION
• Institución: Consejo Nacional de Investigaciones Científicas y Técnicas

Given a positive integer k, a k-dominating set in a graph G is a set of vertices such that every vertex not in the set has at least k neighbors in the set. A total k-dominating set is a set of vertices such that every vertex of the graph has at least k neighbors in the set. The problems of finding the minimum size of a k-dominating, respectively total k-dominating set, in a given graph, are referred to as k-domination, respectively total k-domination. These generalizations of the classical domination and total domination problems are known to be NP-hard in the class of chordal graphs, and, more specifically, even in the classes of split graphs (both problems) and undirected path graphs (in the case of total k-domination). On the other hand, it follows from previous works by Bui-Xuan et al. (2013) [8] and by Belmonte and Vatshelle (2013) [3] that these two families of problems are solvable in time O(|V(G)|3k+4) in the class of interval graphs. We develop faster algorithms for k-domination and total k-domination in the class of proper interval graphs, by means of reduction to a single shortest path computation in a derived directed acyclic graph with O(|V(G)|2k) nodes and O(|V(G)|4k) arcs. We show that a suitable implementation, which avoids constructing all arcs of the digraph, leads to a running time of O(|V(G)|3k). The algorithms are also applicable to the weighted case.
Fil: Chiarelli, Nina. Univerza Na Primorskem. Faculty of Mathematics, Natural Sciences and Information Technologies; Eslovenia
Fil: Hartinger, Tatiana Romina. Cognitiva Etermax Labs; Estados Unidos
Fil: Leoni, Valeria Alejandra. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentina
Fil: Lopez Pujato, María Inés. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentina. Ministerio de Ciencia. Tecnología e Innovación Productiva. Agencia Nacional de Promoción Científica y Tecnológica; Argentina
Fil: Milanič, Martin. Univerza Na Primorskem. Faculty of Mathematics, Natural Sciences and Information Technologies; Eslovenia