TITLE: A "rewiring process" on graphs
QUESTION [8 upvotes]: I am interested in a discrete process defined as follows. We start with a given graph. At each time step we delete an edge $(i,j)$ and add two edges $e$ and $f$; the edge $e$ is incident with $i$ and a neighbour of $j$ (which is not already a neighbour of $i$), and the edge $f$ is incident with $j$ and a neighbour of $i$ (which is not already a neighbour of $j$). A stable configuration is a graph in which a step of the process gives an isomorphic graph. I would like to collect some information about the behaviour of the process and its stable configurations.

REPLY [6 votes]: As Barry Cipra points out in his answer, the rewiring process continues as long as the graph contains some path $hijk$, such that $hj$ and $ik$ are not the edges of the graph. Equivalently, the terminal graphs are exactly the graphs which contain no induced subgraphs isomorphic to the $3$-edge path $P_4$ or to the $4$-cycle $C_4$. Such graphs have been studied before under the name trivially perfect graphs, and the Wikipedia page contains a number of equivalent characterizations of graphs in this class.