Abstract
A graph G is said to be R-perfect if, for all induced subgraphs H of G, the induced regular independence number of each induced subgraph H is equal to its corresponding induced regular cover. Here, the induced regular independence number is the maximum number of vertices in H such that no two belong to the same induced regular subgraph in H, and the induced regular cover of H is the minimum number of induced regular subgraphs in H required to cover the vertex set of H. This article introduces the notion of induced regular perfect graphs or R-perfect graphs through which we study the structural properties of R-perfect graphs and identify a forbidden class of graphs for the same. This further leads to the characterization of R-perfect biconnected graphs. With these results, we derive and prove a general characterization for R-perfect graphs.
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