The quadrangle graph operator

The cycle graph of a graph G is the graph [Formula: see text] whose vertices are the induced cycles of G and where two vertices are adjacent if and only if they are distinct induced cycles that share a common edge. This graph operator was introduced in 1984, and studies on the sequence [Formula: see text] of iterated cycle graphs of G have been made. In 1991, Egawa et al. showed that the sequence of iterated subgraphs can be finite or infinite. In case it is infinite, either the order of the graph in the sequence increases without bound as [Formula: see text], or it is periodic. We call a graph vanishing if the sequence of iterated cycle graphs is finite; expanding if the order of the graph in the sequence increases without bound as [Formula: see text]; and periodic otherwise. The subgraph of [Formula: see text] induced by the 3-cycles in G is called the triangle graph of G, denoted by [Formula: see text]. In 1991, Egawa and Ramos showed that a graph is either vanishing, expanding, or periodic with respect to the triangle graph operator. We define the quadrangle graph of a graph G, denoted by [Formula: see text] to be the graph whose vertices are the quadrangles (induced 4-cycles) in G and where two vertices are adjacent if and only if they are distinct quadrangles sharing a common edge. We show here that a graph is either vanishing, expanding, or periodic with respect to the quadrangle graph operator. We also show by construction that for each positive integer p, there exist quadrangle-periodic graphs with period p.