Transitivity on Minimum Dominating Sets of Paths and Cycles

Abstract

Transitivity on graphs is a concept widely investigated. This suggest to analyze the action of automorphisms on other sets. In this paper, we study the action on the family of γ-sets (minimum dominating sets), the graph is called γ-transitive if given two γ-sets there exists an automorphism which maps one onto the other. We deal with two families: paths Pn and cycles Cn. Their γ-sets are fully characterized and the action of the automorphism group on the family of γ-sets is fully analyzed.

Graph Burning: Mathematical Formulations and Optimal Solutions

“…Although the VKCP and the GBP are different, their approximation algorithms are conceptually similar [7,8,19]. This similarity comes from the fact that the VKCP has a polynomial-time reduction to the minimum dominating set problem, which can be viewed as the problem of burning all vertices in parallel in one single step [19][20][21][22]. Regarding the FP, it aims at protecting vertices from burning given an initial set of fire sources [23][24][25].…”

Free Cells in Hyperspaces of Graphs

“…Graphs have been widely and deeply studied (see [3][4][5][6][7]) and have proved to be an excellent tool for representing and modeling different structures in several areas of discrete mathematics and computation (see [8,9]). As far as hyperspace is concerned, there exist some works relating both subjects.…”