The monophonic number of Cartesian product graphs A.P. SanthakumaranS.V. Ullas Chandran 05C12monophonic pathmonophonic setmonophonic number 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 $J_G[u,v]$ consists of all the vertices lying on some $u$-$v$ monophonic path in $G$. For $S \subseteq V(G)$, the set $J_G[S]$ is the union of all sets $J_G[u,v]$ for $u,v \in S$. A set $S \subseteq V(G)$ is a monophonic set of $G$ if $J_G[S] = V(G)$. 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. Indiana University Mathematics Journal 2012 text pdf 10.1512/iumj.2012.61.4874 10.1512/iumj.2012.61.4874 en Indiana Univ. Math. J. 61 (2012) 849 - 857 state-of-the-art mathematics http://iumj.org/access/