The Monophonic Number of Cartesian Product Graphs

For vertices u, v in a connected graph G, a u-v chordless path in G is a u-v monophonic path. The monophonic closed interval JG[u,v] consists of all the vertices lying on some u-v monophonic path in G. For S ⊆ V(G), the set JG[S] is the union of all sets JG[u,v] for u,v ∈ S. A set S ⊆ V(G) is a monophonic set of G if JG[S] = V(G). The cardinality of a minimum monophonic set of G is the monophonic number of G, denoted by mn(G). In this paper, bounds for the monophonic number of Cartesian product graphs are obtained. Improved bounds and exact values are determined for several classes of Cartesian product graphs. Various realization results are proved.