R-annihilated and independent perfect neighborhood sets in chordal graphs Original Research Article

Let θi(G), ra(G) be the minimum cardinality of, respectively, an independent perfect neighborhood set and an R-annihilated set. We point out some classes of graphs for which the inequality θi(G)⩽ra(G) holds. This study is natural since Favaron and Puech (Discrete Math. 197/198 (1999) 269–284) contains examples of graphs where the difference θi(G)−ra(G) is positive and can be arbitrarily large. We prove that the inequality θi(G)⩽ra(G) holds if the cycles of G satisfy some assertions verified in particular by the chordal graphs. This result generalizes the one concerning trees proved in Cockayne et al. (Discrete Math. 188 (1948) 253–260). We also establish the same inequality for C1,2,2-free graphs, which generalizes the result proved independently in Cockayne and Mynhardt (J. Combin. Math. Combin. Comput., to appear) and Favaron and Puech for claw-free graphs.