Dynamic monopolies and feedback vertex sets in cycle permutation graphs, generalized Petersen graphs and torus cordalis

Abstract

In this paper, we use the connection between dynamic monopolies and feedback vertex sets to establish explicit formulas and new bounds for decycling number $$\nabla (G)$$?(G) when $$G$$G is one of the following classes of graphs: cycle permutation graphs, generalized Petersen graphs, and torus cordalis. In the first part of this paper, we show that if $$G$$G is a cycle permutation graph or a generalized Petersen graph on $$2n$$2n vertices, then $$\nabla (G)=\lceil (n+1)/2\rceil $$?(G)=?(n+1)/2?. These results extend a recent result by Zaker (Discret Math 312:1136---1143, 2012) and partially answer a question of Bau and Beineke (Australas J Comb 25:285---298, 2002). Note that our definition of a generalized Petersen graph is more general than the one used in Zaker (Discret Math 312:1136---1143, 2012). The second and major part of this paper is devoted to proving new upper bounds and exact values on the size of the minimum feedback vertex set and minimum dynamic monopoly for torus cordalis. Our results improve the previous results by Flocchini in 2004.