Identifying codes of corona product graphs

For a vertex $x$ of a graph $G$, let $N_G[x]$ be the set of $x$ with all of its neighbors in $G$. A set $C$ of vertices is an {\em identifying code} of $G$ if the sets $N_G[x]\cap C$ are nonempty and distinct for all vertices $x$. If $G$ admits an identifying code, we say that $G$ is identifiable and denote by $\gamma^{ID}(G)$ the minimum cardinality of an identifying code of $G$. In this paper, we study the identifying code of the corona product $H\odot G$ of graphs $H$ and $G$. We first give a necessary and sufficient condition for the identifiable corona product $H\odot G$, and then express $\gamma^{ID}(H\odot G)$ in terms of $\gamma^{ID}(G)$ and the (total) domination number of $H$. Finally, we compute $\gamma^{ID}(H\odot G)$ for some special graphs $G$

Problems in Domination and Graph Products

The \u27domination chain,\u27\u27 first proved by Cockayne, Hedetniemi, and Miller in 1978, has been the focus of much research. In this work, we continue this study by considering unique realizations of its parameters. We first consider unique minimum dominating sets in Cartesian product graphs. Our attention then turns to unique minimum independent dominating sets in trees, and in some direct product graphs. Next, we consider an extremal graph theory problem and determine the maximum number of edges in a graph having a unique minimum independent dominating set or a unique minimum maximal irredundant set of cardinality two. Finally, we consider a variation of domination, called identifying codes, in the Cartesian product of a complete graph and a path