Maintaining chordal graphs dynamically: Improved upper and lower bounds

Abstract

We study upper and lower bounds for the problem of maintaining a chordal graph G under edge insertions and deletions. Let G be a chordal graph on n vertices and m edges and let (u, v) be the edge to be deleted or inserted. Let k be the size of the maximum clique in G. Our first result is an improved analysis of an earlier approach due to Ibarra [12] to support edge deletions. We can construct a data structure in O(nk2) time such that we can report in O(1) time if G∖ (u,v) is chordal and if it is, we can update the structure in (n+k2) time. We then show using a charging argument that the update time can be improved to O(n2/Δ + k2) amortized time over a sequence of Δ deletions. We develop a data structure to maintain a perfect elimination ordering (PEO) of chordal graphs where we can detect whether G\∖ (u,v) is chordal in O(min{degree(u), degree(v)}) time, and if it is chordal, we can update the structure in O(degree(u)+degree(v)) time. In graphs of bounded degree, our query and update bounds are a constant. Finally, we show that we can obtain a PEO of the graph from a clique-tree in O(n) time after an edge insertion or deletion (against a naive O(m+n) time). This answers a question posed by Ibarra [12]. Regarding lower bounds, we show that any dynamic structure to maintain a chordal graph requires Ω(log n) amortized time per edge addition or deletion or per query to detect chordality, in the cell probe model with word size log n.