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 $O(nk^2)$ time such that we can report in O(1) time if $G{\setminus }{(u,v)}$ is chordal and if it is, we can update the structure in $O(n + k^2)$ time. We then show using a charging argument that the update time can be improved to $O(n^2/\Delta + k^2)$ amortized time over a sequence of $\Delta$ deletions.

• We develop a data structure to maintain a perfect elimination ordering (PEO) of chordal graphs where we can detect whether $G{\setminus }{(u,v)}$ 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 $\varOmega (\log n)$ amortized time per edge addition or deletion or per query to detect chordality, in the cell probe model with word size $\log n$.