Generating cyclic pivot Gray codes for well-ordered $k$-degenerate graphs in constant amortized time

Abstract: A graph $G$ is $k$-degenerate if there exists an ordering $v_1, v_2, \dots, v_n$ of its vertices such that each vertex $v_i$ has at most $k$ neighbors $v_j$ in $G$ with $j &lt; i$. A well-ordered $k$-degenerate graph is a labeled graph on the vertex set ${1, 2, \dots, n}$ in which every vertex $i$ has at most $k$ neighbors among $1, 2, \dots, i-1$. We present the first simple algorithms that generate, rank, and unrank cyclic pivot Gray codes for well-ordered $k$-degenerate graphs, where consecutive graphs differ by the addition, removal, or pivoting of a single edge. Our algorithm generates each well-ordered $k$-degenerate graph in constant amortized time per graph, using $O(n<sup>2)$ space, while ranking and unranking take $O(n<sup>2)$ time and $O(n<sup>2)$ space.