Distance formula for direct-co-direct product in the case of disconnected factors

Abstract

Direct-co-direct product G ⊛ H of graphs G and H is a graph on vertex set V(G) × V(H). Two vertices (g,h) and (g′,h′) are adjacent if gg′ ∈ E(G) and hh′ ∈ E(H) or gg′ ∉ E(G) and hh′ ∉ E(H). We show that if at most one factor of G ⊛ H is connected, then the distance between two vertices of G ⊛ H is bounded by three unless some small number of exceptions. All the exceptions are completely described which yields the distance formula.