On \(\mathcal {C}\) -Perfection of Tensor Product of Graphs

A graph G is \(\mathcal {C}\) -perfect if, for each induced subgraph H in G, the induced cycle independence number of H is equal to its induced cycle covering number. Here, the induced cycle independence number of a graph G is the cardinality of the largest vertex subset of G, whose elements do not share a common induced cycle, and induced cycle covering number is the minimum number of induced cycles in G that covers the vertex set of G. \(\mathcal {C}\) -perfect graphs are characterized as series-parallel graphs that do not contain any induced subdivisions of \(K_{2,3}\) , in literature. They are also isomorphic to the class of graphs that has an \(\mathscr{I}\mathscr{C}\) -tree. In this article, we examine the \(\mathcal {C}\) -perfection of tensor product of graphs, also called direct product or Kronecker product. The structural properties of \(\mathcal {C}\) -perfect tensor product of graphs are studied. Further, a characterization for \(\mathcal {C}\) -perfect tensor product of graphs is obtained.