New insights on GA-H reduced graphs

Abstract Let GA be a hereditary family of graphs and H a family of acyclically directed graphs. A graph G ( V , E ) is a GA-H reduced graph if it can be obtained from a graph GA ( V , D ) ∈ GA by deleting the edges of an edge subgraph H ( V , E ′ ) ∈ H . The present paper complements reference [9] by proving that the following properties are equivalent: Property 1: u → v ∈ E ′ implies N GA [ u ] ⊆ N GA [ v ] . Property 2: u → v ∈ E ′ and v — w ∉ E ∪ E ′ = D implies u — w ∉ E ∪ E ′ = D . This equivalence implies that most of the algorithms in the above reference apply to the GA-H reduced graphs where H is the family of transitive edge subgraphs of the closed neighborhoods containment graphs of the graphs in GA. We also describe a polynomial time algorithm for maximum weight independent set under the weaker condition that H ( V , E ′ ) is a locally transitive edge subgraph of the closed neighborhoods containment graph of GA ( V , D ) .